Most recently that a losing Democratic nominee carried a red county (and vice versa) in every state

Alabama
Trump: 2000 (Barbour, Choctaw, Colbert, Conecuh, Jackson, Lawrence)
Biden: 1996 (Jefferson)

Arizona
Trump: 1988 (Greenlee)
Biden: 2012 (Maricopa)

The most recently that any Trump '20 county other than Greenlee in Arizona voted for a losing Democrat was 1968, when Pinal and Gila also did so (Greenlee being exceptionally small).

Arkansas
Trump: 2004 (Bradley, Clark, Clay, Hempstead, Jackson, Lawrence, Lincoln, Little River, Mississippi, Monroe, Poinsett, Randolph, Woodruff)
Biden: 1892 (Chicot, Crittenden, St Francis)

California
Trump: 1972 (Plumas, Shasta, Sierra)
Biden: 2012 (Butte, Inyo, Nevada, Orange)

Colorado
Trump: 2016 (Alamosa)
Biden: 2008 (Chaffee)

Connecticut
Trump: 2004 (Windham)
Biden: 1992 (Fairfield)

Delaware
Trump: 1880 (Sussex)
Biden: 1992 (Kent)

Florida
Trump: 2004 (Jefferson, Monroe, St Lucie, Volusia)
Biden: 2012 (Duval, Seminole)

Georgia
Trump: 2016 (Burke)
Biden: 2012 (Cobb, Gwinnett, Henry)

Hawaii
Trump: n/a
Biden: 1960 (Hawaii, Kauai, Maui)

Idaho
Trump: 1988 (Clearwater, Lewis, Nez Perce, Shoshone)
Biden: 2012 (Teton)

Illinois
Trump: 2004 (Alexander, Fulton, Henderson, Knox, Madison, Mercer, Putnam, Whiteside)
Biden: 2012 (Kendall, McLean)

Indiana
Trump: 2004 (LaPorte)
Biden: 2012 (Tippecanoe)

Iowa
Trump: 2004 (Benton, Boone, Cerro Gordo, Chickasaw, Clarke, Clayton, Clinton, Des Moines, Dubuque, Fayette, Floyd, Howard, Jackson, Jasper, Jefferson, Jones, Lee, Mitchell, Muscatine, Poweshiek, Tama, Wapello, Webster, Winneshiek, Worth)
Biden: 1976 (Black Hawk, Scott, Story)

Kansas
Trump: 1988 (Crawford, Ellis)
Biden: 2012 (Johnson, Riley, Shawnee)

Kentucky
Trump: 2004 (Bath, Breathitt, Carter, Elliott, Floyd, Knott, Magoffin, Menifee, Pike, Rowan, Wolfe)
Biden: 1992 (Fayette)

Louisiana
Trump: 2004 (Assumption, Pointe Coupée)
Biden: 1992 (East Baton Rouge)

Maine
Trump: 2004 (Androscoggin, Aroostook, Franklin, Oxford, Penobscot, Somerset)
Biden: 1976 (Cumberland, Hancock, Knox, Lincoln, Sagadahoc, Waldo)

Maryland
Trump: 2000 (Somerset)
Biden: 2012 (Anne Arundel, Frederick, Kent, Talbot)

Massachusetts
Trump: n/a
Biden: 1976 (Barnstable, Nantucket)

Michigan
Trump: 2004 (Alger, Arenac, Bay, Gogebic, Isabella, Lake)
Biden: 2012 (Kent, Leelanau)

Minnesota
Trump: 2004 (Beltrami, Big Stone, Chippewa, Fillmore, Freeborn, Itasca, Kittson, Koochiching, Lac qui Parle, Mahnomen, Mower, Norman, Pine, Rice, Swift)
Biden: 1996 (Olmsted)

Mississippi
Trump: 2004 (Benton)
Biden: 2008 (Warren)

Missouri
Trump: 2004 (Ste Genevieve)
Biden: 1976 (St Louis County)

Montana
Trump: 2004 (Roosevelt)
Biden: 2012 (Gallatin)

Nebraska
Trump: 1988 (Dakota, Saline, Thurston)
Biden: 2012 (Douglas, Lancaster)

Nevada
Trump: 1968 (Mineral, White Pine)
Biden: 1996 (Washoe)

New Hampshire
Trump: 2004 (Coös)
Biden: 2012 (Carroll, Rockingham)

New Jersey
Trump: 2000 (Monmouth, Salem)
Biden: 2012 (Morris)

New Mexico
Trump: 2000 (Colfax)
Biden: 1996 (Los Alamos)

New York
Trump: 2004 (Franklin, Niagara, St Lawrence, Suffolk)
Biden: 1992 (Clinton, Columbia, Dutchess, Essex, Saratoga)

North Carolina
Trump: 2016 (Scotland)
Biden: 2012 (New Hanover)

North Dakota
Trump: 2004 (Benson, Steele)
Biden: 1948 (Sioux)

Ohio
Trump: 2016 (Lorain, Mahoning)
Biden: 1996 (Hamilton)

Oklahoma
Trump: 2000 (Cherokee, Choctaw, Haskell, Hughes, Latimer, McIntosh, Muskogee, Okmulgee, Ottawa)
Biden: n/a

Oregon
Trump: 2004 (Columbia)
Biden: 2012 (Deschutes, Marion)

Pennsylvania
Trump: 2004 (Beaver, Fayette, Luzerne, Washington)
Biden: 2012 (Chester)

Rhode Island
Trump: n/a
Biden: 1960 (Washington)

South Carolina
Trump: 2016 (Clarendon, Dillon)
Biden: 1996 (Charleston)

South Dakota
Trump: 2004 (Corson, Day, Roberts)
Biden: 1992 (Ziebach)

Tennessee
Trump: 2004 (Benton, Clay, Grundy, Hardeman, Houston, Humphreys, Jackson, Lake, Lauderdale, Overton, Perry, Smith, Stewart, Trousdale, Van Buren)
Biden: 1964 (Haywood)

Texas
Trump: 2016 (Frio, Jim Wells, Kenedy, Kleberg, La Salle, Reeves, Val Verde, Zapata)
Biden: 2012 (Fort Bend, Hays, Tarrant, Williamson)

Utah
Trump: 1988 (Carbon)
Biden: 2012 (Grand, Salt Lake, Summit)

Vermont
Trump: never (Trump carried one county, Essex County, which has never voted for a losing Democrat)
Biden: 1976 (Addison, Bennington, Caledonia, Chittenden, Franklin, Grand Isle, Lamoille, Orange, Orleans, Rutland, Washington, Windham, Windsor)

Virginia
Trump: 2004 (Buchanan, Covington City, Dickenson, Nelson)
Biden: 2012 (Chesterfield, James City County, Lynchburg City, Montgomery, Stafford, Virginia Beach)

Washington
Trump: 2004 (Cowlitz, Grays Harbor, Mason, Pacific)
Biden: 2012 (Whitman)

West Virginia
Trump: 2004 (Boone, Braxton, Brooke, Fayette, Logan, McDowell, Marion, Mingo, Webster)
Biden: n/a

Wisconsin
Trump: 2004 (Adams, Buffalo, Crawford, Dunn, Grant, Iron, Kenosha, Lafayette, Pepin, Pierce, Price, Trempealeau, Vernon
Biden: 1992 (Door)

Wyoming
Trump: 1968 (Carbon, Sweetwater)
Biden: 2012 (Albany)

This is kind of a blunt measure of how recently a given party has made inroads in the other party's strength in a state. (It's blunt because, of course, on the one hand, inroads can be made without flipping counties, and on the other hand, some counties are so small that flipping them is not necessarily a sign of much that is significant, especially if they are not only small but isolated.) And of course, it is worth keeping in mind that Delaware, Hawaii, Connecticut, and Rhode Island are considered statistically few-countied.

Nevertheless, it's worth noting two extreme outliers, both of which are also the states that hold the record for longest streaks of loyalty for the two parties: Vermont and Arkansas. (Arkansas' streak of voting Democratic post-Reconstruction lasted longer than Georgia's.) It wouldn't be surprising, however, to see a Democrat carry Washington County, Arkansas (which voted for Trump in 2020) in the near future. It wouldn't be terribly surprising to see a Republican carry Orleans County, Vermont (which voted for Hillary Clinton in 2016) in the near future either.

Amongst large states, California is also something of an outlier for the Republicans. The only other states in which Trump carried no county carried by a losing Democrat any less recently than 1988 are Delaware, Hawaii, Nevada, Massachusetts, Vermont, and Wyoming. Of course, if he had pulled off the near-flip in Stanislaus, this index would have improved dramatically for him (as the most recently Stanislaus had voted for a losing Democrat was 2016).

Interestingly, as of 2000, the large state where this index would have painted a particularly dire picture for Republicans would have been New York:

Alabama
Bush: 1988 (Cherokee, Walker)
Gore: 1992 (Montgomery)

Arizona
Bush: 1988 (Greenlee)
Gore: 1976 (Coconino, Pima, Santa Cruz)

Arkansas
Bush: 1988 (Conway, Fulton)
Gore: 1964 (Ashley, Drew)

California
Bush: 1972 (Plumas, Shasta, Sierra)
Gore: 1976 (Contra Costa, Imperial, Marin, Monterey, Napa, San Benito, San Mateo, Santa Barbara, Santa Clara, Sonoma)

Colorado
Bush: 1988 (Bent, Conejos)
Gore: 1976 (Boulder, Lake, Pitkin, Saguache, Summit)

Connecticut
Bush: n/a
Gore: 1992 (Fairfield)

Delaware
Bush: 1924 (Kent)
Gore: 1932 (New Castle)

Florida
Bush: 1980 (Baker, Bradford, Calhoun, Columbia, Dixie, Franklin, Gilchrist, Glades, Gulf, Hamilton, Jackson, Lafayette, Levy, Liberty, Madison, Okeechobee, Putnam, Sumter, Suwannee, Taylor, Union, Wakulla)
Gore: 1996 (Orange)

Georgia
Bush: 1988 (Crawford, Greene, Marion, Pulaski, Taylor)
Gore: 1964 (Baker, Bibb, Burke, Calhoun, Chattahoochee, Clay, Clayton, DeKalb, Dooly, Dougherty, Jefferson, Mitchell, Muscogee, Peach, Quitman, Randolph, Richmond, Stewart, Talbot, Telfair, Terrell, Twiggs, Warren, Washington, Webster, Wilkinson)

Hawaii
Bush: n/a
Gore: 1960 (Hawaii, Kauai, Maui)

Idaho
Bush: 1988 (Clearwater, Latah, Lewis, Nez Perce, Shoshone)
Gore: 1976 (Blaine)

Illinois
Bush: 1988 (Cass, Christian, Hancock, Jefferson, Jersey, Pike, Randolph, Saline, Vermilion, Williamson)
Gore: 1976 (Champaign, Fulton, Henderson, Henry, Knox, LaSalle, Mercer, Peoria, Putnam, Whiteside)

Indiana
Bush: 1988 (Sullivan)
Gore: 1976 (LaPorte, St Joseph)

Iowa
Bush: 1988 (Adair, Adams, Appanoose, Audubon, Buena Vista, Calhoun, Carroll, Cherokee, Clay, Crawford, Dallas, Davis, Decatur, Delaware, Emmet, Fayette, Franklin, Guthrie, Hamilton, Hancock, Hardin, Humboldt, Iowa, Jasper, Keokuk, Kossuth, Lucas, Madison, Marion, Marshall, Monona, Monroe, Palo Alto, Pocahontas, Poweshiek, Ringgold, Sac, Taylor, Union, Warren, Washington, Wayne, Winneshiek, Woodbury, Wright)
Gore: 1976 (Black Hawk, Buchanan, Cedar, Clayton, Clinton, Jones, Louisa, Muscatine, Scott, Story, Winnebago)

Kansas
Bush: 1988 (Crawford, Ellis)
Gore: 1976 (Douglas)

Kentucky
Bush: 1988 (Bath, Carlisle, Carroll, Carter, Gallatin, Graves, Greenup, Henry, Hickman, Livingston, Lyon, Magoffin, Marion, Marshall, McCracken, McLean, Menifee, Morgan, Owen, Robertson, Trimble, Union, Webster, Wolfe)
Gore: 1976 (Jefferson)

Louisiana
Bush: 1988 (Acadia, Allen, Ascension, Calcasieu, Cameron, DeSoto, East Feliciana, Evangeline, Jefferson Davis, St Martin, Vermilion, West Feliciana)
Gore: 1976 (Caddo)

Maine
Bush: 1968 (Penobscot, Piscataquis, Washington)
Gore: 1976 (Aroostook, Cumberland, Franklin, Hancock, Knox, Oxford, Sagadahoc)

Maryland
Bush: 1980 (Kent)
Gore: 1996 (Charles)

Massachusetts
Bush: n/a
Gore: 1976 (Barnstable, Nantucket)

Michigan
Bush: 1988 (Alger, Baraga, Delta, Keweenaw, Ontonagon, Schoolcraft)
Gore: 1992 (Macomb, Oakland)

Minnesota
Bush: 1988 (Anoka, Beltrami, Blue Earth, Chippewa, Chisago, Clay, Clearwater, Cook, Dakota, Grant, Isanti, Jackson, Kanabec, Kandiyohi, Kittson, Koochiching, Mahnomen, Marshall, Mille Lacs, Murray, Nobles, Norman, Pennington, Polk, Pope, Red Lake, Renville, Stevens, Traverse, Washington, Yellow Medicine)
Gore: 1976 (Fillmore)

Mississippi
Bush: 1980 (Alcorn, Amite, Calhoun, Chickasaw, Clarke, Copiah, Franklin, Grenada, Itawamba, Jasper, Lafayette, Leake, Lee, Madison, Marion, Monroe, Montgomery, Pike, Pontotoc, Prentiss, Tate, Tippah, Tishomingo, Union, Walthall, Winston, Yazoo)
Gore: 1992 (Hinds)

Missouri
Bush: 1988 (Audrain, Chariton, Clark, Clinton, DeKalb, Dunklin, Gentry, Howard, Iron, Knox, Lewis, Linn, Marion, Mercer, Monroe, Nodaway, Oregon, Pike, Ralls, Randolph, Reynolds, St Francois, Shannon, Shelby, Worth)
Gore: 1976 (St Louis County)

Montana
Bush: 1988 (Blaine, Hill, Lewis and Clark, Lincoln, Mineral, Missoula, Rosebud)
Gore: 1976 (Glacier)

Nebraska
Bush: 1988 (Dakota, Saline, Thurston)
Gore: n/a

Nevada
Bush: 1968 (Mineral, White Pine)
Gore: never (Gore carried only Clark County, which had never voted for a losing Republican)

New Hampshire
Bush: 1968 (Coös, Hillsborough)
Gore: 1976 (Cheshire, Grafton, Merrimack, Strafford)

New Jersey
Bush: 1908 (Hunterdon, Sussex, Warren)
Gore: 1992 (Bergen, Monmouth, Passaic, Salem)

New Mexico
Bush: 1956 (Eddy, Lea)
Gore: 1976 (Bernalillo, Doña Ana)

New York
Bush: 1908 (Schoharie)
Gore: 1992 (Clinton, Columbia, Staten Island, Suffolk)

North Carolina
Bush: 1988 (Caswell, Duplin, Greene, Haywood, Hyde, Jones, Swain)
Gore: 1892 (Chowan, Granville, Hertford, Northampton, Pasquotank, Tyrrell, Vance, Warren, Washington)

North Dakota
Bush: 1988 (Benson, Divide, Mountrail, Nelson, Ransom, Sargent, Steele, Towner)
Gore: 1948 (Sioux)

Ohio
Bush: 1988 (Columbiana, Harrison)
Gore: 1992 (Franklin)

Oklahoma
Bush: 1988 (Atoka, Bryan, Caddo, Coal, Cotton, Craig, Garvin, Greer, Harmon, Jefferson, Johnston, Kiowa, Love, McCurtain, Marshall, Mayes, Murray, Nowata, Okfuskee, Osage, Pittsburg, Pushmataha, Seminole, Tillman)
Gore: 1960 (Cherokee, Haskell, Hughes, McIntosh, Muskogee, Ottawa)

Oregon
Bush: 1988 (Coos, Tillamook, Wasco)
Gore: 1976 (Benton, Hood River, Washington)

Pennsylvania
Bush: 1988 (Armstrong, Clinton, Indiana, Westmoreland)
Gore: 1976 (Bucks, Delaware, Lehigh, Montgomery)

Rhode Island
Bush: n/a
Gore: 1960 (Washington)

South Carolina
Bush: 1984 (Edgefield)
Gore: 1964 (Allendale, Bamberg, Clarendon, Hampton, Jasper, Lee, McCormick, Marion, Orangeburg, Richland, Williamsburg)

South Dakota
Bush: 1988 (Aurora, Brown, Brule, Charles Mix, Corson, Day, Lake, Marshall, Miner, Minnehaha, Moody, Roberts, Spink, Union, Ziebach)
Gore: 1976 (Clay, Dewey)

Tennessee
Bush: 1984 (Lincoln)
Gore: 1964 (Crockett, Haywood)

Texas
Bush: 1988 (Bastrop, Baylor, Briscoe, Burleson, Caldwell, Camp, Cass, Clay, Comanche, Concho, Cottle, Crosby, Delta, Dickens, Falls, Fannin, Fisher, Foard, Franklin, Galveston, Hall, Hardeman, Hardin, Haskell, Hudspeth, Jasper, Karnes, Kent, Kleberg, Knox, Lee, Limestone, McCulloch, Marion, Menard, Milam, Mitchell, Montague, Navarro, Nolan, Nueces, Orange, Polk, Rains, Red River, Sabine, San Augustine, San Jacinto, San Patricio, San Saba, Shelby, Stonewall, Swisher, Terrell, Throckmorton, Titus, Travis, Trinity, Tyler, Waller)
Gore: 1944 (Kenedy) (Excluding Kenedy County, which cast 228 votes in 2000, the Gore counties in Texas that had most recently voted for a losing Republican were Maverick and Zapata, which had last done so in 1916; they cast 9,229 and 2,616 votes in 2000, respectively.)

Utah
Bush: 1988 (Carbon)
Gore: n/a 

Vermont
Bush: never (Bush carried four counties, none of which had ever voted for a losing Democrat)
Gore: 1976 (Addison, Bennington, Chittenden, Franklin, Grand Isle, Lamoille, Rutland, Washington, Windham, Windsor)

Virginia
Bush: 1988 (Lee, Tazewell, Wise)
Gore: 1992 (Emporia City, Newport News City)

Washington
Bush: 1988 (Asotin, Clark, Kittitas, Klickitat, Pend Oreille, Skamania, Wahkiakum, Whatcom)
Gore: 1976 (King, San Juan)

West Virginia
Bush: 1988 (Barbour, Calhoun, Clay, Gilmer, Greenbrier, Hancock, Marshall, Mason, Monongalia, Nicholas, Pocahontas, Raleigh, Randolph, Summers, Taylor, Tucker, Wayne, Wetzel)
Gore: 1960 (Kanawha)

Wisconsin
Bush: 1988 (Barron, Burnett, Chippewa, Clark, Forest, Iron, Kewaunee, Lincoln, Manitowoc, Marathon, Polk, Price, Racine, Rusk, St Croix, Washburn)
Gore: 1976 (Columbia, Grant, Green, La Crosse, Lafayette, Rock, Sauk, Vernon)

Wyoming
Bush: 1968 (Carbon, Sweetwater)
Gore: n/a