Just to add something of a correction. The seamen pressed by the Royal Navy were British subjects, not Americans. The Navy had the legal right to press British seaman who did not possess a certificate of exemption, such as those engaged in fishing or whaling. British seamen serving on American ships were not exempt and were, therefore subject to being pressed from the American ships that they were serving on. As British seaman, therefore, they would have continued in their pressed role on Navy vessels, until such time as they were paid off when whichever vessel they were serving on ended its commission. They would then be free to sign on any vessel they wished to, but would still be subject to the press. However, as the Treaty of Fontainebleau had ended the Napoleonic Wars, at least temporarily, long before the Treaty of Ghent, it meant that with peace with America having been established as well as peace with France, that the Navy could be reduced, leading to massive decommissioning of Navy vessels, and the releasing from service of thousands of pressed seamen. Those pressed from American ships would simply have joined this enormous pool of unemployed British seamen.

Donald Hickey, The War of 1812:A Forgotten Conflict is, I believe, one of the standard works on the war.

The war was ended on the basis of status quo ante bellum, meaning nothing changed - all captured lands were returned, all prisoners and seized material was given back, and all freed slaves taken by the British were to be returned (they ended up making an indemnity payment instead). Notably, Britain refused to recognize American maritime rights, though they carefully avoided impressing men from any American ships when Napoleon escaped and forced them to resume the war in Europe. After Napoleon there was no compelling reason to do so, so it never came up again.

Hickey doesn't mention the fate of impressed sailors specifically, however status quo ante suggests that they were consigned to their fate as British seamen.