For me, it went something like this:
Baby delivered, showed to me and told it was a boy (first time) and a girl (second time). Put onto my stomach for me to hold, and have some skin to skin, and the cord clamped, DH cut the cord. I said oh my god about 400 times. And hello to baby. Then one of the team took the baby and put it on the warmer while they did checks, weight, apgar etc, and DH was invited to come and dress baby. They don't bath/wash baby, although they did wipe DS off a bit because he was covered in meconium. DD was just wiped off a little bit, she was born a lot cleaner. They don't bath them because the gunk they come out with is good for their skin.
Not too sure what was really happening at that point though because I was busy delivering the placenta. They gave me back the gas and air, and then did a local anasthetic so they could stitch me up. Somewhere in there they asked about the vit k injection and I consented, so they did that, and then they cleaned up the bed around me, gave me the baby, and we did a feed, took pictures rested a bit etc. For my first one the anaesthatist had to come and remove the epidural as well, and make sure I was recovering ok from that.
My first baby, they bundled me into the shower about half an hour after that happened, because the delivery suite was busy, then packed us off to the ward. The second time they left us there for a good hour, popping in occasionally to check on us, and then sending me into the shower before we went down to the ward.