Have you tried a well-women's clinic, Anne? They might be able to fit you in better than the GP. Depo is a hormonal-based contraceptive, so if you have problems w/BCPs talk it over w/a nurse practitioner at a well-women's clinic. Tubal ligation is also provided on the NHS, although the wait list for vasectomy is shorter.
The place I've been waiting to get into is a family planning clinic - my GP doesn't do coils or Depo.... I've only had one issue with BCPs (vaginal dryness that was bad enough to make sex miserable) and only had it over here (in the US I was on low-estrogen pills and they were totally fine but they don't seem to have those over here....) I'd be willing to try Depo once to find out if it had the same effect, I think....
I don't want to tie my tubes 'cause as much as we don't want kids, I don't think either of us is willing to say absolutely never. I doubt we'll change our mind, but we've also both agreed that "we're going to try to avoid it, but if it happened, it wouldn't be the end of the world." I know that sounds sort of lame, but... well, we are sort of lame. Heehee.
Oh, and geally, at the risk of TMI, the clinic says the period thing with the coil is because I haven't had kids so I'm not very, er, stretched out and when your period is about to end, it's a bit slippery which makes things easier....
She said otherwise it would be hard to get in and quite painful....