Joined my first ship on my 17th Birthday, went to sea the following day! I think the skipper got paid extra as he was my legal guardian, not sure how true that bit was though. However, this was in 1984!
The average application time is 12 to 24 months. Plus you will have 10 weeks at Raleigh followed by specialist training of a few months followed then by Submarine training, so even after you've joined you won't get on your first submarines for a good 12 months. So I would guess you are going to be over 18 by the time you get to sea.