Nappy's plan was to swarm the British Navy in numbers. He could lay down about 3 ships for every 2 the British could, since he had the resources of a continent. While his starting navy wasn't as large or competent as the British one, given enough time he'd be able win and invade Britain. This requires two assumptions: 1) the British are incompetent and give France the time to build up its navy and 2) the continent plays ball and stays under France's thumb. Both are highly unlikely, but vaguely possible. So we have that.
Getting America to join? That's a bit harder. France needs to end impressment, and Britain needs to have kept it going. Given the time required for France to build up its navy, America would've already lost the War of 1812. I doubt we would've been eager for revenge quite so soon. So unlikely you could really get America to join in...
UNSPEAKABLE SEA MAMMAL ALERT!!!!!
*facepalm*
Under certain conditions, Britain could certainly be invaded. If for whatever reason, America and Britain went to war today, you'd see Americans storming the beaches of Sussex. A millennium ago, the Normans Sealioned England. If you want to bring up Sealion, keep it to Nazi Germany ca 1942.