The article I read just said that the Queen has visited Harry and Meghan at Frogmore, but it didn't give a date. I think it might have been before William and Kate visited over Easter weekend.
Harry does seem relaxed for a dad who could get the call any moment to say his wife's in labour with their first child, but I can't imagine he'd do engagements between the birth and an official announcement.
If Megan wanted him to do it that way I’m sure he would.
*my snarky opinion only*