Had elements of a natural CS with my second ELCS.
Delayed cord clamping. They waited maybe two minutes, with the surgeon/assistant holding my baby curled up on her hands - he wasn't crying, so no distress at waiting for cuddles. The surgeon asked me before he cut whether I was happy we'd waited long enough, and told me the cord had finished pulsing. He'd have waited longer if I asked.
Immediate skin to skin and constant parental presence. Once cord was cut my baby was moved straight to my chest (pooing a sea of meconium on the surgeon, the floor and my chest on the way!). Baby stayed there maybe 20 mins (during which I attempted a feed but couldn't get the angle right as I was tilted head down and onto left side and there was a sea of meconium on us!) before being briefly taken for weighing etc - DH went with baby. This was the first time my baby cried and it turns out it was because they have injected vit K at that point, despite birth plan being covered in request for oral. Separate issue, but still quite angry about that. Baby was then brought back to my bedside and held by DH, though could have gone back on my chest if I'd asked. DH then carried baby from theatre to recovery, which I was surprised at, but it just happened that way.
If you don't ask, you don't get! But I would ask for specific things you would like rather than just for a 'natural CS', as the term means different things to different people. It is likely your hospital will agree to some things but not others, and it'd be sad to miss the things you can have because you can't have them all.
Also, ask early. An if possible, as to see a senior midwife in the hospital team rather than community team to write a CS birth plan together. She will know what has been done before, and which surgeons will be open to what you're asking for. I planned my CS birth as part of a VBAC clinic to plan a VBAC birth - the NHS had invited me to meet lovely senior midwife so she could pursuade me to VBAC, but whilst we did pull together a just-in-case-you-change-your-mind VBAC birth plan, she also helped me plan a CS as I was sure that's what I wanted. Your community midwife might be able to arrange for you to meet with a suitable hospital midwife, though I would start with a 'I was wondering if this type of birth night be possible' approach rather than making any demands - there's time for demands later if softly softly doesn't get you there 
Good luck!