This should have all been discussed and settled before the marriage!
It depends on the family and there are many variables.
How do the children feel; are all the children on the same page?
How does the bio/dad feel?
Which dad will be supporting or help supporting them?
If they keep bio-dad's name, does the new stepfather threaten the marriage?
If the stepfather causes misery in the family or leaves over this, I say good riddance, but I know that's hard to do.