Re the closeness of the brothers...
BIL told his brother, your husband, last week about the pregnancy.
So why weren’t you posting last week “AIBU - they didn’t tell us she was 39 weeks pregnant!”
If the birth so soon after has come as a surprise, sounds like your husband didn’t even ask his brother about the due date, or how far along.
Although your BIL has made no effort to come to you (with a reason they feel is a good one, and I agree) the way you word it is that it’s your husband who is refusing to go to them. You’d be happy to.
So your husband won’t go to them, he has a chip on his shoulder about deference due to his age (maybe???), and he doesn’t bother to ask about the dates for this pregnancy.
It sounds like your BIL & SIL might just be fed up with him.
I do think it’s all a shame, but it doesn’t sound like it’s come from nowhere 