A city could have sprung up where Washington, DC sits called Georgetown, like the posh neighborhood in the same today.
I always imagined *Washington had a viable shot at being a capital for a Dominion of America
anyway due to a planned city being an excellent compromise idea for north and south, independent or not, and the area being as good as any for a middling site. In which case I can see *Washington Proper being considered an expansion of Georgetown or the new planned downtown or so forth.
Basing on if America stayed British...
Minneapolis -
Albion (briefly once its official name)
Lexington -
Lancaster (proposed alongside York, but then news came of the Battle of Lexington)
Austin -
Waterloo (its name before changing to honor the Father of Texas)
Columbus -
Franklinton (Franklin being a big name anyway, and the myth of Columbus being far less strong to a British America)
Detroit -
Lernoult (maybe. The
fort was renamed to that, but the village kept the name Detroit and may or may not have switched in time as Americans moved in/the 1805 fire still happens)
St. Louis -
St. Lewis or
Pancore (called as such by British stationed in the area per the French locals' nickname of Paincourt for it, but American Francophilia kept the formal name)
Louisville - perhaps after King George of some sort or
Whitehome (or some corruption. The site's very first, colloquial name when civilians came in was The White Home)
Denver -
Golden (Denver for twenty-four hours was named "Golden City" for obvious miner reasons before the renaming to honor the governor, butterflies and many cities slicing off "City" as they grew would lead to this)
San Francisco -
Drakesport (San Francisco Bay was called even in American maps straight up to 1846, Sir Francis Drake's Bay or Sir Francis Drake's Port - shortened to just the last two words, easy to slam into one word per names like Shreveport, LA or Williamsport, PA)
Knoxville -
Amherst (one of the few times I'd cop to a parallel name than an older name as Knoxville didn't quite have one and Amherst was Secretary of War at the same time Henry Knox was, and the Americans liked Amherst for his role in the French and Indian War)
Columbia, SC -
Granby (same deal as Washington vis-a-vis its now-annexed neighborhood Georgetown)