I have had 2 c sections. One as a private patient at public hospital and one in the private hospital. Both times my husband was with me through out (though at the public hospital they wouldn't let him be with me while they did the spinal/epi). Once baby was checked over (which only took a few minutes) he was placed on my chest while the surgery was finished. When they moved me to recovery my hubby held bub and once we were there I had bub back on me. Hubby was by my side the whole time.
My recovery first time round was harder as I had a PPH and required several blood transfusions. I wasn't allowed out of bed for a few days which made it more painful getting up the first time. Second time round I was up out of bed the next day and made sure to move often. It was a much quicker recovery and within 3 weeks I was back to all my normal duties including driving and lifting my toddler.