And if you change your pronouns, Bertie Wooster will never fuck them up because he barely has room in his brain for one set of them per person. As soon as you tell him, the old ones just evaporate. He might ask Jeeves about it later, but it'd be to the tune of "I say, Jeeves, why didn't you tell me that Bingo was a woman this whole time? I've been calling her a bloke for years; she must think I'm a perfect ass."
To be clear, he doesn't understand that she transitioned. He thinks that she's always been a lady. He'd try to explain it to someone and accidentally be the most supportive ally.
an aunt, probably: What's all this nonsense about young Bingo, then? I hear he's gotten it into his head that he's a woman. Going about in dresses and such.
Bertie: Oh, I was confused as well, but it turned out to be rather a large misunderstanding. Bingo is a woman, always has been.
Aunt: That can't be right, Bertie; he was at Eton with you, you absolute chump.
Bertie: Well, yes. Some sort of scholarship program, perhaps? I'm fuzzy on the details. But she's very definitely a woman. She told me so herself, and I daresay she would know. Bit embarrassing for all of us, really; we mistook her for a bloke for years, the poor girl. She must have been too polite to say anything about it.
Aunt: But he's gone his whole life up until last week looking like a man! If he were a woman, why would he not present himself as such?
Bertie: There was a dress code. I don't know how many times I was told off for a scruffy tie.
Aunt: I don't mean at school, you dunce. Even if - and it's still nonsense, mind you - even if I were to accept that Eton somehow permitted this ridiculous state of affairs, what about afterwards?
Bertie: Oh, I haven't the foggiest. I've long since given up on explaining the fairer sex, as well you know.
Aunt: Bertram, he was christened 'Richard'.
Bertie: Yes, bit of an odd choice on her parents' part. I mean, you don't see many girls named Richard, what? I say, do you suppose that's why she goes by 'Bingo'? If I were a lady saddled with Bingo's Christian name, I should likely choose something else too.
Aunt: Have you spoken to Jeeves about all this?
Bertie: Naturally.
Aunt: And? What is his evaluation?
Bertie: He says that when a young lady asserts that she is, in fact, a lady, one ought to take her word for it.
Bertie: Very sensible, I thought. One can always trust Jeeves in these matters.
Bertie: Say, when's lunch?
I don't know that the canonical Bertie Wooster could be called "progressive" (or "politically engaged" or "aware of anything that's going on outside of his immediate sphere of acquaintances with funny nicknames") but you can't argue he wouldn't support gay marriage. Bertie Wooster neither likes nor understands straight marriage, but he fights for his friends who inexplicably want to do that.