I've been induced twice. Both times the induction didn't start until a good 18 hours after I'd got to the hospital, and the baby didn't actually arrive until late the next day. I wasn't then wasn't out of hospital for another five days, first time because dd ended up in scbu, second time because ds struggled to feed.
I don't have a stepchild, but if your DSS was my own child, my main concern would be making sure there was someone lined up to look after him that weekend in case I was still in hospital/giving birth. The last thing you want is to be still in labour and discover his mother has booked a weekend in Glasgow.
When DS was born, my DM came to look after DD and ended up staying a week. I was really grateful she did, as it meant DD had someone who put her needs first and she felt loved and cared for rather than second-place, as might have happened with me and the baby in hospital and DH taken up with our needs.
If you want DSS to feel loved and valued and an important part of your family, there's an argument that you want to make his first weekend with you and the new baby as smooth as possible, and for your DH to have as much headspace to care for him as possible. This may be easier the weekend after.
I think it depends a bit on how old he is though tbh. A teenager is a different scenario to a three year old.