the thing is it kind of depends on the day and time you give birth.
At min you usually get tea and toast and a cuddle/feed with the baby, then you sometimes get moved on to the post-natal ward if they need the delivery room.
If it's daylight hours your DH/P will usually be allowed to stay. If it's nightime they will often be asked to leave which can be a bit sad.
What happened when I had my first baby -
I delivered at 1am. I had an injection to deliver the placenta and had a brief cuddle with the baby, then I was stitched up which seemed to take ages.
I had another, better cuddle with the baby and then they put a nappy on him and I tried to feed skin to skin, but he wasn't interested so after half an hour or so more they put him in the clothes we'd brought and swaddled him in a hospital blanket as he was getting a bit cold.
I was offered tea and toast, and had a shower and changed my clothes while DP cuddled DS.
Then we would probably have been moved, but the labour ward was quite quiet, so DP and I were left alone with our new DS and allowed to bond for a couple of hours. I tried to feed a few more times but no luck.
At about 4 am they needed the room so I was wheeled downstairs to the antenatal ward. At that point DP was supposed to go home but they let him stay to "settle me in" - in fact what happened was he conked out on the bed with me and we all three slept!
At about 6am someone came round and said he'd have to leave as the other ladies were waking up and they didn't want to upset anyone out of visiting hours.
DP left, went home, and came back at about 10am which was when visiting hours started.
We didn't start texting until the next morning as we didn't think anyone would be awake at 1am.
I delivered my next baby at home (bliss!) but if I had to do it again, the one thing I would try to do differently is feed BEFORE they stitched me up. DS was a bugger to get to latch, and I strongly believe that was due to a too-long delay in trying to feed.
If I had another hospital birth I would insist on a physiological third stage and use the time to feed the baby.
HTH.