Um, the war of 1812 was fought concurrently with the height, decline and fall of Napoleon's Empire, that's why so little attention was paid to it by the British, and why whatever happens, the best generals and the ones who have already shown talent (such as Wellington who had already made something of a name for himself in India) are going to be sent to Europe to fight the French. Despite the cultural importance of the War of 1812 for the US, the British were basically uninterested, and only burnt down Washington as a reciprical event for the US burning down York (roughly modern Toronto).
As for that bit about Quebec, I'm not sure what you're going at as it was British (so they're not taking French exiles in), while the Royals are not going to want to be six months away from Europe. If you mean the French holding onto Quebec until the War of 1812, the likelyhood of there even being a War of 1812, or indeed a United States to fight in it, are very much reduced.