See, it may just be me, but I'm of the opinion that it's just not their business. If I was intending to throw a bit New Years Eve party or something where people will be there until 3-4am I wouldn't send them a letter. And my guess is that would be much louder than a birth. Most people are asleep at 3am anyways so probably won't even notice people coming to your place. Not to mention if you had a baby say in hospital would you send them a letter saying you now have a baby? Because those cries at 2am could be far louder than anything you may come out with it.
Now if you get on with your neighbours and talk to them regularly then yea I would let them know, but all I know is in my complex if I were to say anything all it would do is get people's backs up and start complaining about things that aren't even their business.