There have been quite a few threads on this topic, so you might find those worthwhile to read (if you can avoid the unfortunate slide to bickering many fall into - guilty as charged).
From my understanding, Britain had no direct interest in territorial gains, as in, they had no interest in conquering territory directly, in such a way as to start a war. That isn't to say however that they wouldn't take territory as a form of reparation, or indirectly take territory (i.e. setting up a protectorate).
Now for the war in 1812, one of the biggest obstacles to either side in the war was the territorial familiarity of the other side. Whenever either side went on the offensive, because the other side had good positions, it almost always failed.
Where one side had the advantage, then it shifted - i.e. The Great Lakes, America had better bases, which gave them a great advantage in defence, whilst I don't think Britain had anything equivalent, but the did have fantastic shipbuilding abilities. On the flip side, Britain DOMINATED the sea, which was one of the reasons that cities like Baltimore, and DC were vulnerable, and that the naval war was where Britain was stronger.
With the Great Lakes, essentially they could push to have the USA respect (in treaty) a border for the Confederacy (not that one, the Indian one) - assuming it survived in any meaningful manner - if Britain signed the treaty, they may have a stake in ensuring it did survive.
This would mean that if the USA crossed the border in any manner military or administrative, without acceptable reason, that the British would have a legitimate cause for war, and were signed on to do so.
This does NOT mean however that settlers will respect this, which is where it gets messy. In order to ensure that settlers didn't just settle the territory, you'd need forts, and people to man them - and the Confederacy just couldn't. This (IMO) would mean that Britain would end up fortifying the frontier in exchange for the Confederacy allowing British colonists in, or in exchange for taxing the American settlers - which would lead to messy legal issues for both the USA and Britain, as the USA shouldn't have troops or lawmen there, in a treaty signed by law, but these settlers have American citizenship and may petition for help. A sensible settlement would be that the USA goes "Dude, you're not in my jurisdiction. You live on their land, you play be their rules". By the way, send that lovely food over here, we'll not tax you, and look at these lovely goods you can buy off us on the cheap!
This messiness means that the USA couldn't actively expand westwards as a government policy, but the settlers could - if that isn't suitably handled by the Confederacy, or the British, then the USA at some point could well take advantage of a weak period, or the settlers could, and any polities that emerge could join the USA, undoing the Confederacy.
Now shifting the war causes very lengthy discussions on the details typically - but my two cents have been that if there wasn't a freak storm to save DC, then that could cause some serious damage to the US in terms of morale, and in order for Britain to try and ask for anything beyond Maine, or reparations, they NEED to win on the Great Lakes, or successfully invade the South. I put my money on the lakes. Anything else, you need a very different war, with Britain being free to exert its full might on North America.
TL;DR The British need to win the lakes, if so, a Protectorate, with some complications could lead to Britain being the primary "authority" west of the Appalachians, besides Spain. (Not coloniser, it'll probably rely on free-settlers, which will be expected to at least pay lip service, if thye do anything of the sort). Anything more and IMO, they'd need a spectacular victory, that would require significant military investment to gain anything further.