Hadn't the Americans promised, during negotiations for the Treaty of Ghent, to cede parts of Maine north of the Penobscot to British Canada? I swear I remember reading that somewhere, I just can't remember where.
You're right. This goes back to the Revolutionary War and carried through 1812. The Maine/New Brunswick border dispute goes back deep into the French Acadia days. That said, settling a preexisting border dispute post-war by ceding land is quite different from the wholesale unraveling of a state admitted to the Union. I'm not saying it couldn't happen, I'm just saying that failure of the US in the War of 1812 would need to be ENORMOUS the consequences of such for internal US politics would be catastrophic.
For the list that I had made, everything is the same until President Madison is killed in August 1814. The British might be able to argue that America's admission of the State of Louisiana was illegitimate since the sale of Louisiana to America was illegitimate. With the US in utter disarray, and economically broken, they don't really have a choice but to give the British the territory.
I'd imagine that it's more likely than not that disunion lies in the US's future going forward. Why trust the government in Washington with defending the nation in the future when it failed so miserably that a co-equal state was absorbed by a foreign power?
I'm trying to conceive of the economic and geopolitical circumstances that would lead Britain to favor going all-in on the American continent and what expense that would have for empire building in Asia and Oceania. What returns is Britain getting from its new territory? A new agricultural sector for cotton, sugar, timber, etc. A new link for trade with Native Americans, a geographically closer destination for sending settlers than Australia, potentially access to the copper resources around Lake Superior.
What costs? Needing troops to contain the United States, needing to deal with settlers via the US, playing diplomatic tango between former Spanish colonial possessions and the Spanish Caribbean.
I don't think it would immediately affect India operations; the BEIC was largely independent and succeeded in subjugating the Marathas in the late-1810s without large infusions of support from the Empire. It could have an effect on Anglo-Burmese relations into the 1820s and beyond. OTL, the First Anglo-Burmese War nearly bankrupted British India. ITTL if the Empire is also expending resources to securing and developing North America, difficult choices will need to be made with regard to Britain's expansion in Asia versus North America. Likewise, development in Australia likely slows to a crawl. I think that settlers and convicts who go to Australia OTL are likely to be diverted to British North America, although the more isolated Australian continent may be a more attractive destination for convicts than the diplomatically complicated Gulf region.