The decision to have your child(ren) at your birth should ideally be always accommodated. Unfortunately, some institutions (and also many caregivers) are reluctant to facilitate this choice. This may be compared to the resistance of caregivers to allow fathers into the labour room before the 1970s'. 'They were of no use, and only got in the way!' The push to make birth a more human experience has led to nearly all partners now being present at the birth, and the increasing trend to include siblings.
Some places require that the parents write a letter of request to the hospital administrator. This is merely a formality, and one you should comply with if you want to assert your choice. It is a formal notification that is rarely denied, but gives you another 'hurdle' to jump, leaving the choice for only the determined. There may be conditions attached though, such as having an extra support person (which is advisable anyway), or taking the children out in the event of complications. You should still have the right to accept (or question this) if you feel it is not reasonable.
Some birthplaces place limits on the number of people who are able to support a woman in labour. There is no medical reason for imposing this restriction, and one that should be challenged if you want to accommodate your child (or yourself for that matter).
It should be part of every birthplace to facilitate an area, or an environment, that is child friendly and considers their needs. Many birth centres have toy boxes and areas for children to play. But this is the exception for most birthplaces, rather than the norm. The hospital should be aware of providing a safe environment, with hazards out of reach (needles and chemicals) and areas where the child can play (or sleep). You may need to bring a small mat, or bed, for the child if they need to sleep.
Hopefully in the years to come, children at births will be as accepted as fathers are today.