As longtime fans of The Simpsons know, the Battle of New Orleans, whose bicentennial we celebrate today, occurred two weeks after the War of 1812 ended. This turns out to be one of things we know that just ain't so: Donald Hickey points out in Don't Give Up the Ship (U. of Illinois, 2006) that while British and American commissioners signed a peace treaty in December 1814, it did not go into effect until both governments legally ratified it in February, so the war legally continued for several weeks after Andrew Jackson's lopsided* victory. This observation raises an intriguing counterfactual question: if the British had captured New Orleans in 1815, would Britain have been able to retain the city, and a chunk of the Louisiana Purchase, as legitimate prizes of war? After all, the United Kingdom had not recognized the legitimacy of France's sale of Louisiana to the United States. The question becomes more interesting when we realize how close Britain came to capturing the vital seaport: as David and Jeanne Heidler recently observed**, though British troops failed to take New Orleans from the south, another army planned to move on the Crescent City from the less defensible north, via Mobile and Baton Rouge. British General John Lambert captured Mobile's harbor defenses in February, just before the war ended, and he and his colleagues would have stood a good chance of occupying New Orleans if the war had lasted a few more months.
I doubt, however, that Britain would have kept New Orleans for long. The Treaty of Ghent specifically restored the status quo ante bellum, denying Britain a legal right of conquest to Louisiana, and even if British officials decided to deny the validity of the Louisiana Purchase, doing so would have obliged them to return New Orleans to its previous European owners, the French. I doubt they would have found this an attractive alternative. More importantly, Whitehall's primary objective during and after the War of 1812 was the defense of its existing North American colonies, Upper and Lower Canada. It had authorized the attack on the Gulf Coast as part of a diversionary campaign to draw American forces away from the vulnerable Canadian provinces. Holding territory in the Gulf region would have interfered with Whitehall's post-war efforts to secure the U.S. - Canadian border through diplomacy (e.g. the Rush-Bagot agreement that partially demilitarized the Great Lakes). New Orleans might have provided Britain with a bargaining chip in future negotiations with the Americans, but I've seen no evidence that Foreign Office ministers were thinking this way in 1815.
I don't mean to suggest, though, that a hypothetical British capture and occupation of New Orleans would have had no consequences at all. But we need to move away from geopolitics and the American master narrative of war and expansion to determine those consequences. We should follow the lead of Alan Taylor, Gene Smith, and Nathaniel Millett, all of whom have drawn our attention to a previously under-studied aspect of the War of 1812: the decision by thousands of African-American slaves to treat the British Army as a liberation force, and to flee to the protection of their lines. Millett observes that Edward Nicolls (no relation) recruited runaway slaves into a British volunteer force after Britain seized Pensacola in 1814. In Florida hundreds of those freedmen subsequently formed an autonomous maroon community (the “Negro Fort”) which stood until American troops destroyed it in 1816; others took refuge with the Seminoles. Nearby Louisiana had a large (35,000) and restive slave population in 1815, and it is very likely that Louisiana slaves, by the hundreds if not the thousands, would have responded to a British occupation of New Orleans by seeking refuge with the invaders. How this would have altered the “big picture” of American history I know not, but consider: Louisiana had its own maroon communities in the late eighteenth century, and in 1811 had generated the largest slave revolt in North American history, the German Coast uprising. The state already had a culture of slave defiance, and it is probable that a slave exodus to British-occupied New Orleans would have strengthened this culture in the 1810s and '20s. In any case, abscondance and the possibility of liberation would have changed and improved the lives of hundreds of people, and that is as worthy of comment as speculation about the impact of a British victory on American national power. History is about people, not just nation-states.
* About 50 Americans were killed and wounded, versus more than 2,000 British casualties.
** "'Where All Behave Well:' Fort Bowyer and the War on the Gulf, 1814-1815," in Tohopeka: Rethinking the Creek War and the War of 1812, ed. Kathryn E. Holland Braund (U. of Alabama Press, 2012), pp. 182-199.