Look for a "try a tri" event - they generally let you ride anything that's roadworthy - I've seen someone compete on a BMX bike before. Mountain bikes are not uncommon in Novice events - you wont look out of place. Check with the race organiser or if its a British Triathlon Federation sponsored event check the rules on their website.
Hope you enjoy your first tri.
Would probably be worth fitting some slick tyres, other than that a mountain bike will be fine. There are often people riding mountain bikes in sprint triathlons
You can use any bike, but my top tip is that you always run off the bike after every ride you do. If you are using cleated bike shoes, leave some trainers outside your house and run straight away, even if it's just 200m as its this part of the triathlon that always catches newbies out. And invest in elastic laces for your running shoes. Good luck.
I just did my first Sprint Triathlon at Dorney Lake. I loved it and massively recommend it to start. I have started a blog and an instagram page. Just for fun but also to help people like you and me!
p.s. 'grakay' - I agree, the run did catch me out a bit. Legs felt very odd for the first 250m. Good tip.