What if a biological male is assigned female gender at birth? Would they be transgender, or just female?

That's part of the reason that "biological male" isn't a highly informative/technically-precise term, but assuming you mean something like Swyer syndrome, there isn't really a reason to worry about estrogen making them infertile, so in context using the phrase "transgender women" would have been clear regardless.

I think, since the assignment is typically done by a medical professional (at least for recording a birth certificate), the assumption is that the gender is either a quickly fixed clerical error or reflective of a phenotype that presents as sufficiently male/female to convince a professional.

"Convincing a professional" is a reasonable proxy for biological sex, given that the cases that would fool a medical professional are in many (most?) cases sufficiently ambiguous that a reasonable person could take either side for what the biological sex is.

1. When is that going to happen? (Genuinely curious if it does happen)

2. It is really up to the person.

Regarding 1:

https://www.bbc.com/news/magazine-34290981

They would continue to be male, forever. "Female" is a sex, not merely a gender identity.

At birth, nobody is "assigning" genders to you, they are attempting to observe your sex from primary sex characteristics. If they see vulva, they are observing a female; if they see penis/scrotum, they are observing a male.

While there are edge cases, it is generally pretty simple.

This is a bit off-topic, but "at birth" is a seriously wrong phrasing - they were assigned that phenotypical gender well before birth.