An interesting question indeed.
I would imagine it could have been tough for some of the British soldiers, sailing across the Atlantic then ending up in a new land far from home. If some had any sympathies for the American cause it could have been easy to fine a new “home” with the Continental Army, or otherwise just blend into the civilian population.
That said, it would probably have been tough to go back home to England, if they were listed as a deserter. It makes me wonder if mercenaries (Hessians, etc.) might have had similar experienced.
If I had to guess, there were likely more Continental army deserters, though likely more from the Biritish ranks than we might think.