Two VBs for me, and and I was up on my feet not long after the births, so capable of doing whatever I needed to bub-wise. So no, DH didn't stay either time.
I actually had a huge room with a double bed when I had DS1, so DH spent a fair bit of time with us in the evening (we had snuggles in bed etc), but he left late evening each night. We have two pampered dogs that needed him equally as much as I did LOL. With DS2, I only had a standard hospital bed, but I didn't need DH to stay - all was going well, I was fine and so was bub. And DS1 needed him at home. I can't imagine things will be any different with this next bub.