Haven't read the whole thread so sorry if this has been said already but I have recently done the whole speedy passport for a new baby thing...
You can get a passport on the 7 day express service if you make an appointment at a passport office e.g. Newport, Peterborough, Liverpool. You MUST make this appointment in advance and you pay when you book. You do not get a refund if you have to miss the appointment. It costs about £90. I have just done this for my newborn and the passport actually came in just a few days.
HOWEVER - you MUST have the full birth certificate, which means you must have registered the baby's birth. This has to be done within 42 days anyway (fewer in Scotland I think), but where I live there were no appointments for about 25 days!! So it would have been impossible for me to get a passport for my son in the timeframe you are aiming for. There are sometimes cancellations but that would mean checking every day and then hoping you could get a passport appointment immediately.
I would also bear in mind that you might not be in a fit state yourself, though of course you might be absolutely fine. No way to tell in advance unfortunately! I would have been fine to travel with my first pretty much immediately but not my second (EMCS).
I moved to the other side of the world when my first was only 11 weeks old and it was fine, and I am taking a toddler and a 7 week old on my own to visit family next week, which will involve 2 flights. Perhaps I am mad! If I was physically well I wouls be fine travelling with a very recent newborn but passports etc make it very difficult.
One other thing to consider - your baby will not have had his first jabs yet, which for some parents would be a dealbreaker.
In general terms though I would say that flying/travelling is MUCH easier with a tiny than with a toddler!