I'm not sure what you'll need from first sittings of higher- but you'll need about AAB at advanced higher, similar to the A-level required grades. If you're intending to take three advanced highers and you're predicted good grades, give it a shot.
Be warned though- a lot of people have been commenting on the quality of its undergrad teaching though, PhD students seem to get lumped with a lot of it (in order to let the senior staff do research instead), and student satisfaction is pretty low. They might be rectifying this, but its something to consider before applying. In addition, London living costs are much higher, and you'll have to pay fees to study, so it will be much more expensive than staying in Scotland. It might be worth doing the sums to see if its worth the several thousand per year more it'll be than going to one of a number of very good Scottish universities.