If you go over, baby will potentially be only 4 weeks old. C-section or any other complications, infections etc, or anything wrong with baby and you won't be able/ wanting to travel that soon.
Vaccines can be brought forward a little bit - though not ideal, but if you give birth early you could probably just about squeeze the first round in.
Passport and registering baby might be tight. We had a month to wait for our appointment to register our second child. Then had to apply for passport once we had the birth certificate. So that wasn't back within your 6 week limit (we weren't travelling!).
First baby we got registered, passport back and flights booked within 7 weeks. That wasn't for a holiday though. It was essential travel.
If your flights are booked, can you just add an infant close up? I'm not sure of how it works with booking flights for nameless babies with no passport or birthdate info! Would you have to wait til baby is born before adding them to your ticket? And getting visa etc.
Flying with a baby is really easy. They sleep a lot. Breastfeeding is MUCH easier than formula when travelling as you can't run out of breastmilk in the instance of delays etc. Get a sling if you do go.
Honestly, although it is manageable, I'd delay the trip if you can. Even by a few weeks. There are so many chances for things to go not to plan, it's not worth the risk in my view. It would be sad to be there and be so sore that you couldn't walk around etc.
But if you do go, plan well and enjoy!