IME (3 babies) - I got out of the water before the birth part all 3 times.
Baby is out which is very intense - they pass baby straight up to your arms, and put a towel over the top of them to keep them warm. Sometimes they are totally wrapped in the towel, but I think 2/3 times I had proper skin to skin with the towel just laid on top. Much ohhhhh hello and loveliness. They don't do anything at this point, unless there is something very wrong e.g. baby not breathing (I think in a c-section also they have to take the baby instantly for oxygen etc). They are taking apgar score, but you don't really notice.
It's probably about 20-30 minutes that you just have the baby in your arms and nothing else seems to be happening, staff may be going in and out of the room, then they will look for whether the placenta is coming. I didn't have the injection for mine, and it took between 20-40 mins after birth for it to come out. I just knelt up and did a tiny little push and it all came out - feels like no trouble at all after the head and shoulders parts!
I requested all 3 times for the cord to stop pulsating before being clamped and cut and was able to have this.
At the point where the placenta is coming, or before this if someone else (e.g. dad) wants to have a hold they generally wrap the baby up properly like a tiny little bundle so it doesn't get cold.
Once placenta is out, they will examine you to see if you need any stitches. You can hold the baby during this, or dad/birth partner can or there may be a fishbowl type cot in the room to put the baby down.
At this point with DS2 they felt he was still a bit blue, so they asked to take him for some further observations, which I agreed to but asked DH to go with him, which he did. Babies are normally slightly blue/purple when they are born but it's a problem if they don't pink up quickly enough. They were giving him oxygen but he wasn't really responding properly to it/breathing on his own. So I can't remember if they brought him back to me or they wheeled me to him, but they basically didn't want him out of the heat lamp/oxygen thing for too long because he wasn't breathing effectively. He had to go to SCBU for a couple of days. (He was fine and able to come home after a week).
But generally if everything is fine with baby and everything is OK with you they will leave you to it for a bit, as you no longer need urgent attention, and come back when they have free staff. You will probably be offered something to eat and drink, and after this you might be encouraged to go for a bath/shower especially if you have bled a lot. It was probably a couple of hours each time from the moment of birth to being taken to a ward. They usually push you in a wheelchair rather than expecting you to walk (but this may depend on the hospital/birth!)
Baby is normally dressed ready to go to the ward, not usually before. In the UK IME they do not take the baby away to do the newborn anatomy check, this is done at your bedside. I had the younger two in Germany and they did take the baby away to the other side of the room. They found something a bit unusual with DS3 (nothing hugely problematic) so they kept him there for a bit longer while they found a paediatric urologist to come and look at him.
DH/birth partners generally have to leave once you've been moved to the ward unless you happen to be moved during visiting hours.