This was the sort of the issue behind my question. In the short term the presence of a British garrison in New Orleans might affect the balance of forces available. In the longer term the possibility of a Britain holding the Mississippi basin, including the Ohio and Missouri etc has the potential for significant impact. These rivers provide a way to spread colonisation efforts into the interior much faster than they ever could in Australia.I would think it would not affect the outcome on the eastern seaboard, but it very well could effect exactly where the US western boundary is fixed. UK power projection into Mississippi, Illinois, western tennessee, western Kentucky and Wisconsin May be stronger than US.
If a British Louisiana Territory (BLT - sorry, I couldn't resist it!) survives, it would significantly affect the development of the West Coast and of Canada. If in this ATL Britain still loses the 13 colonies, but holds on to BLT there would be no Compromise of 1850, no Annexation of California, major changes to Texas and the SW, no Alaska purchase. Development of the railways would almost certainly take a different pattern, with the transcontinental links taking much longer to appear.
Politically it would be interesting too. Britain would probably still have Canada plus effectively the Mississippi Basin. On the East Coast would be a truncated USA. It's the West Coast that gets most complicated. Russia is still active in Alaska and spreading down the Pacific Coast from the North. I don't know if there were any issues between Britain and Russia over this in OTL, but I suspect in this one, there would be. British influence on the West Coast would almost certainly spread much further south into Washington State and Oregon, while Spanish/Mexican control would be spreading North. Where a Britain/Canada heading south meets Spain/Mexico heading north is something to think about, and the impact of that contact on relationships back in Europe.
There is also the possibility of an attempt by France to make a comeback somehow.