Even directly after the Revolutionary War, relations with Britain were relatively amicable. British emigrants saw the U.S. as a likely choice as their future home because they spoke the same language and shared a similar culture. During the terms of Washington and Adams, American citizens might have viewed British nationals with some distrust, but at that point the lineages and languages were so similar it would've been nearly impossible to notice.

Think of it this way - in the precolonial era, the Thirteen Colonies and Britain traded vital resources back and forth for the continued success of both parties. After American independence, the demand for these resources from either side hadn't gone away at all. As such, it was the best option for both countries - more specifically, merchants and merchant fleets within those countries - to resume trading operations. These waves of trade helped repair relations between Britain, America, and their respective citizens - until that trade was cut off by Britain during the Napoleonic Wars, at which point the War of 1812 happened.

The only thing which turned British immigrants away from the U.S. was Canada. Canada, like the U.S., had massive undeveloped forests, farms, and fisheries waiting for opportunistic individuals to capture them. More importantly, Canada was still a British colony, meaning that those resources could go directly towards the Crown and its Empire. Canada was a valid alternative to the U.S. as far as emigration from the Isles was concerned, and both places received a steady flow of British immigrants.