There was the Chesapeake and Ohio Canal. But there was a reason it was never completed OTL, it's a lot harder to build a canal there than across Central and Western NY. Even the stretch along the Potomac is arguably more difficult than the stretch of the Erie Canal that paralleled the Mohawk. If you look at a topo map you can see why.
But setting that aside I would have the same questions about how the whole western frontier would shape up in such a scenario. It seems more like a counter-factual than an Alt History POD. For example Western NY would be a hard position for the US to hold without also controlling the Mohawk valley or Oswego. Seems like a the British would almost certainly take Fort Niagara which would bookend WNY. American supply lines would probably have to follow the Cherry Valley Turnpike but at that point it went no further west than Cazenovia IIRC. But if the British take WNY then why not, say Erie PA with its harbor, where do you draw the line. Maybe a more reasonable frontier would be the Black River then across the Adirondacks to the south end of Lake Champlain. The British neutralize Sacketts Harbor and take Fort Ticonderoga to guard the approaches to Canada. But that still leaves the issue of Michigan and the old Northwest.
So I think the US connections to the interior would really be dictated, at least to some extent, by what territory the US still held in the Northwest. But I suppose if we just go with it and if we assume that the US somehow holds together after a crushing defeat and the loss of such strategic territory the Ohio and Cumberland is the logical direction to take. It may become enough of a strategic imperative that it gets built with great difficulty (compared to OTL Erie Canal). But then the question becomes one of funding and control. If the economic imperative isn't enough to fund it's much higher cost through private investment, and the individual states won't do it (its path crosses several), would it form an early public work of the Federal government? Such a move would almost certainly be extremely controversial for an era when the Federal government simply didn't do that sort of thing. But again as a strategic imperative to secure the Ohio from the British perhaps they push it through.
So the implications there are huge, the Federal government builds and then controls the primary access point to the west. It sets the precedent that the Feds can and should take on huge public works and creates new methods of funding (do they issue new Federal Canal Bonds against future toll revenue, get creative on taxation etc) it also makes the Federal capital of Washington DC the outlet to all that commerce. So DC will rival Philadelphia pretty quickly, truly a phoenix from the ashes of the War of 1812.
Meanwhile Pittsburgh combines its OTL position with that of OTL Buffalo NY as the western terminus of the Canal and quickly becomes America's second city. This probably alters the whole canal system in Ohio. The Ohio and Erie Canal may not be built for the same reason that people were at first against the Oswego Canal, as they feared a canal connection to the north would drain commerce away to Canada. So Cleveland probably doesn't develop as well either.
