Callum,
you can only apply for one base initially but you will be asked for base preferences in your interview and you can mention your flexibility then. I would apply for LGW initially as it's the bigger base and expanding faster so chances are that you'll get an assessment day sooner.
It takes a few month between your interview (your interview, not when you send off your application!) and starting to fly.
First you have to pass the interview and it takes about one week before you hear back.
Then you have to go through the whole referencing and CRC process which can take quite a while depending on your history but will take at least 4 weeks due to the CRC.
And then they have to place you on a training course. That all depends on when the next place is available and may take a while.
And then you have to go through the course (about 3 weeks) but once you've passed your course you'll go flying pretty much straight away.
So if it goes as smoothly as it possible could and everything goes in your favour you are still looking at a minimum of two month between interview and flying. And that would be super quick, and half a miracle to be honest.
You do get paid from the first day of your training course though, not from your first day of flying.
Good luck with your application!