ABird, hello!
IIRC we actually jumped in the pool wearing our lifejackets. I can't seem to remember having been in the pool without the jacket on.
Makmo, if you're worried though, don't be. The trainers will help you as will your fellow trainees. The swimming thing isn't that big a deal really, as on my course there were a couple of people who absolutely couldn't swim but they were fine. People just looked out for them.
Good luck.