[...] Is there a difference between American and British English? [...]
Judging by the above posts (all AmE), I'd say YES, there does appear to be a difference. "Bye-bye" is one of the many variants often heard in BrE (and I've heard it used in AusE, NZE, IndE, IrE) — but I've never heard it associated with being childish, gay, effeminate or patronising, nor being used essentially by grandmothers, nor used specifically to toddlers or infants.
I wonder whether the status of the expression in AmE has changed in recent times: in the past there have been loads of songs from American songwriters with "Bye-bye" in the titles or lyrics (and I don't mean kiddy songs).
So, mysina, it seems you should think twice before using it to Americans, but I'd say you'd be safe using it in most other parts of the English-speaking world.
Ws👁 Smile :)