I used to do it while at Uni and it was fantastic fun. Mainly did smaller club meets at Thruxton (closest circuit), but have also done BTCC and Brit GT a few times. Even had the pleasure of being on post at Brands with the chap who rugby tackled that nutter at the British GP years ago!
If it appeals, and from your post clearly it does, then as others have said, get in touch with your local(ish) marshalling club and sign up and start doing it. You certainly won’t be at the British GP in your first year, you have to do a certain number of meetings and be good at it to do it. I think most are BRDC as well (which isn’t an issue really – you can still join the BRDC). The big events are good fun as the cars etc are impressive, but the most fun ones are the club events – much friendlier atmosphere and racing is generally better.
Definitely give it a shot!