No requirements for a private flight.
But it used to be the case that a flight plan to France got rejected if you didn't tick the box for jackets.
I'm not sure how that worked, thinking about it, as that information is not part of the FPL message that gets transmitted.
I would suggest that lifejackets (worn, not under the seat) is a minimum recommendation for an SEP channel crossing. Crossing the Solent, I wouldn't bother.