I'm a paediatrician, don't attend deliveries any more but have attended several hundred in my time! As everyone has said, we are called at the last minute (usually - every now and then I would be the first one called to an emergency section, as in before the obs or anaesthetic teams and turn up to an empty theatre without even a mum in it yet - this did often cause theatrical sighing, watch looking at and ranting!!) and do not interfere at all with the actual delivery/cutting cord. Often if the baby comes out pink and screaming I won't even hang around to examine him or her, the only reason I would want the cord cut straight away would be if the baby was not breathing and needed resuscitation or was extremely premature and we needed to keep them warm (there is a benefit to delayed cord clamping in extreme preterm babies but there's also an increase in the mortality rate if they get cold so unless there's a reliable way to keep them warm til the cord's stopped pulsating (and that would include us being able to do the required resuscitaton) the risk/benefit ratio falls on the side of cutting the cord early).
Paediatricians are aware of the benefits of delayed cord clamping, even if they have not kept up to date with research it is taught on the neonatal resus course which is compulsory for us to do every few years.
In the case of the OP's baby, if I was called to the delivery, I would come at the last minute, stay out of the way but cast an eye as soon as the baby is born, if he or she looks in good condition I would be very happy and in fact would encourage delayed cord clamping and skin to skin. I would however want to do a quick examination at some point so once the cord was cut and you were happy for me to do so, I would take the baby to the resuscitaire which will be in the delivery room, look at colour and work of breathing which would take a couple of minutes, then give baby back, tell you everything looks ok and leave. Obviously if the baby needed breathing support or anything else we would do whatever was needed and in that case we may have to take the baby away a bit quicker and spend a bit more time doing stuff. However in my experience this is very rarely needed in cleft lip/palate babies as long as they don't also have a very small chin (ie Pierre-Robin syndrome).
When i need to take a baby to NICU I will, if at all possible, make sure mum and dad have had a cuddle beforehand, if the baby's too sick for that I will make sure they have at least seen baby before I go.
OP, if, when you come in in labour, you ask the midwife if you can speak to the paeds doc on call, it is part of our job to come and speak to prospective parents who may/will need NICU admission and you will probably speak to the actual person who will come to your delivery rather than a consultant in clinic who almost never attends deliveries - you and they can decide how much or little intervention you want as long as things are going to plan (and that may even be not having a paed in the room but having them called as an emergency if there are problems)