The closest thing I can find to what you want in the UK is the Atlantis Submarine Voyage at Legoland Windsor. Here you travel in a vehicle with a view underwater. However, it may disapoint you as: It is in a tank, not the sea; part of the vehicle always remains above water; the vehicle is propelled along by the track, it is not self-locomoting.
Equally, if you are interested in travelling outside of the UK, then there are many opportunities. See for example this article on escapeartists, or closer to the UK than most of those, this canary island based company. I recall seeing several firms offering subs of varying sizes in the Cancun area of Mexico on a recent (2015) trip.
This article indicates a company was planning in 2002 to run tours out of the UK south coast, but it seems that the tours never happened.
Finally, see this question.