Games development is a dead industry. Loads of people are interested in it, but in the UK there are so few places that do it most developers are unemployed. Furthermore, it's widely known to be a pretty shocking job even if you're at a top studio.
BTECs are not recognised by most good universities - do A-levels, they'll prepare you better.
For computer science, you basically want Maths, and any other couple (most people seem to do Physics and FM, since lots of universities also like Further Maths). BTECs are too vocational - they don't prepare you at all for a lot of the theory.
Having said that, I'm not sure what sort of uni you'd be applying for...