If the British win in New Orleans, they will control New Orleans. And they will have just violated a treaty of peace signed a month ago, after spending two decades at war. In the eyes of Europe and the British public, continuing the war would be a bad move.
If the British decide to demand anything, the Americans can say 'no'. What power projection do the British have over US territory outside of southern Louisiana? New Orleans is important, but it's also the middle of winter, so the Americans in the Mississippi valley won't be missing out on much.
Even if the Spanish get Louisiana back, the area will probably end up containing so many Americans that it'll end up back in their hands. The Spanish can't do much to keep the Americans (or anyone, for that matter) out.
If the British decide to demand anything, the Americans can say 'no'. What power projection do the British have over US territory outside of southern Louisiana? New Orleans is important, but it's also the middle of winter, so the Americans in the Mississippi valley won't be missing out on much.
Even if the Spanish get Louisiana back, the area will probably end up containing so many Americans that it'll end up back in their hands. The Spanish can't do much to keep the Americans (or anyone, for that matter) out.