My first c-section was an emergency, bub was in distress and there was meconium so she was taken to the table for paediatrics to check her but she was all good so thry brought her to me. I had a cuddle while being stitched but then she went with DF and I went to recovery as my BP was very low and I was shaking a lot so I needed to rest a little (it was 2.30am so I was ok with this).
My second was planned so when he came our they were bringing him to me but he started to pee everywhere, lol... they quickly put a nappy on him and then he came straight to me and I breastfed him while being stitched up. It was awkward as I was lying down and had wires and stuff in the way but it was nice to have that skin to skin contact. He then stayed with me in recovery and all the way to the ward.
I think it really depends on the hospital as they all have their own procedures - best to ask them at your next visit and let them know your wishes in advance if you have to go down that path.