Raleigh recently hosted the Plymouth area mountain bike championships so there is a team, maybe a Navy team and you used to be able to sign out mountain bikes as well. As Tommo said, speak to PTI's when you get to Raleigh. You'll meet them soon enough as they canvass all new entry classes for sports people to see if anyone can represent the Navy.
There is a RN/RM cycling association. It can be found here, http://www.navycycling.co.uk/
They cover most disciplines in cycling and I've been told that one of PTIs at Raleigh is heavily involved in the cycling scene. There will be loads of opportunities to go cycling once you've left Raleigh.
As you can see from the cycling association website, there is indeed the opportunity to throw yourself downhill at speed on two wheels. As anyone who's gone down one of the black ski runs on Bavarian Surprise! Once you've finished training there is also increasing opportunities to gain mountain biking qualifications. I've recently done a MIAS Level 2 leaders award through the pusser, at no cost to myself.
At the moment mountain biking is not considered Adventurous Training, it's a Challenging Activity, however that may be changing. If it becomes AT it will be easier to organise and gain leader/instructor qualifications.
You want to speak to PTI Sargent Godell at HMS Raleigh, He's RAF but a real nice guy and very approachable. He handles all the Adventurous Training, I went paragliding a few weeks ago for 7 days in Wales all free of charge. But he takes the Mountain Biking classes every week at mandatory PT in phase 2.