I've always wondered about this. I have a hard time believing Mary would have delivered Him herself. I think we can eliminate St. Joseph, as that wouldn't really have been considered "men's work" at the time.
And then I suppose it depends which account of the gospels you want to go on. given the two different versions - I suspect either "the innkeeper's wife" [I would think this would be a skill an innkeeper's wife might well have had, given the numerous traveller's and situations such a person might have encountered] OR perhaps, Mary's kinswoman, Elizabeth? If Mary had gone to assist Elizabeth )and I'm not saying she delivered John, either) perhaps Elizabeth returned the favor, coming in her time of need to help out?
How privileged that unknown person was to have been the first besides the BVM to touch the "Word made flesh."