Not sure how wheelchair friendly the embassy is. I know there weren't ways to get the pushchair around, so I had to get it up several sets of stairs. There must be a way if you have a wheelchair, but I didn't see it. Maybe you can call and ask about provisions for wheelchairs (who knows, maybe they have to meet with you somewhere else and then you won't have to wait as long
). Some of the tube stations are wheelchair friendly, but many aren't as well. I'm not sure about how you get around there--it might be easier to take a cab from Paddington, which is wheelchair friendly.
I didn't take my British husband with me if that helps any. I did have to get a letter signed by my husband and a solicitor friend certified it for us. (It says notarized, but the certified stamp our solicitor used was just fine). That was for the passport bit--saying it was okay with him to give her a passport.
I also did it as an overnight trip from Cardiff (not too far from you) since I made an appointment in the morning, but I have in-laws in London that I could stay with. You'd probably have to get up fairly early to do it all in one day, but the trains from Bristol to London are pretty frequent.
Also, keep checking on appointment times. I got one a few days out from my appointment. It may have been a cancelled appointment. Don't stress about it all though unless you have to have a passport soon. I waited quite a while because it was difficult for us to get down to London, and we didn't have any problems getting it all done still.