Like the OP, i live away from family (2787 km, by google maps). When my sister had her son, I offered to fly over and meet the chap. I would’ve arrived about 3 or 4 days after the due date. Luckily, the flights were cheap. She declined as she reckoned that she’d be knackered and had enough people fussing.
The next time that i saw them, was on Christmas day when she walked into the pub carrying the kid. All the family and local villagers were there. As i plan to be a thoroughly irresponsible (fun) uncle, it seemed like a good way to first meet him!
Enjoy your holiday and when you’re back, go see the kid.