Hospitals might vary but my Consultant said that if I'd agree to the spinal, the process assuming healthy baby would go like this:
Baby gets lifted out, drape can be lowered a bit or they can hold them up high (for some reason most of our obstetricians are six foot plus men).
Baby goes on your feet/legs for a 1 minute or so whilst the cord pulses, before being cut. Anesthetist or student or midwife can take photos for you.
If everything seems okay and you want that, baby comes to mum for skin to skin although they recommend nappy & hat because apparently getting meconium out of your hair is not fun straight after surgery but that can be done next to you or on you.
If there are issues, baby would be checked over, given nappy & hat & blanket if you want and then brought over to you.
Ideally baby not to leave you from within minutes of birth through recovery to the ward and then home.
I had a category 1 emcs with ds, didn't know what day it was, my blood pressure dropped very low when they got him out and once they'd checked him over, they wanted me to hold him, apparently it can help stabilise you so I think they really push it in most hospitals these days. In the end dh held him and then put him in the bed with me to go to recovery and he stayed with me until he needed to go to NICU (for reasons relating to my labour not the section).
Seeing the most senior consultant anesthetist tomorrow so that should be fun.
On section recovery, the first time the health visitor came around to see ds, I was hoovering. Think it was day 11 or so, she pretty much exploded at dh and I haven't hoovered since (ds is 3... him and dh have matching dysons and do all the hoovering, it's great).