Honestly? There’s not a companion I don’t see Bill getting along well with.
She and Rose would be good, supportive friends, swapping stories about their families and lives and probably hanging out together in their pjs eating ice cream.
Bill would really respect Martha, and Martha would be only too happy to explain the science of things to Bill. They would geek out about scifi and Harry Potter together. When the reality of all the deaths she’s witnessed got too much for Bill, she could depend on Martha to listen and understand, since she too knows the cost of traveling with the Doctor.
Donna would immediately adopt Bill as her younger sister and the two of them would make the Doctor’s life hell, being constantly sarcastic to him and to each other.
Once Jack realized that Bill was gay, they would probably do the thing where they flirt in a ridiculously over-the-top, not at all serious way all the time. They would laugh a lot together and make jokes about straight people.
Amy’s never really had a girl friend before, so Bill could be Amy’s first real introduction to sleepovers and going to the pub with friends and all those normal things that Amy is so separated from, and in return, Amy would tell Bill all the stories that she has in her head, the ones she remembers as well as the ones she’s making up.
Bill and Rory both have the ability to see situations clearly and not get overwhelmed by the glamour of it, so they would probably really appreciate having someone else who’s straightforward and who just gets it.
Bill and Clara would definitely have not-so-subtle crushes on each other. Their dynamic would be all about flirty banter while saving the world.
I think River would tend towards a more maternal or professorial role. Bill would be kind of weirded out by the fact that the Doctor has a wife at first, but once she got over that, she would think River was super awesome, and River would be absolutely impressed by Bill’s humor and wit.
“What killed it?” “Time. Just time. Everything’s dying now. All the great civilizations have gone. This isn’t just night. All the stars have burned up and faded away into nothing.” “They must have an atmospheric shell. We should be frozen to death.” “Well, Martha and I, maybe. Not so sure about you, Jack.”