I echo the passport concerns. You cannot fast track a first passport. I had 3 weeks from the time dd was born until we flew internationally, after much hassle, expense and four trips to London, we got it the day before we left.
My advice- book an appointment in advance. They offer a 7 day service, but it can take 12 days to get an appointment in London. Then you call every 15 minutes and try to get a cancellation, but each call takes 5 minutes. An absolute nightmare with a 2 day old baby, ( and a husband overseas). It might not be so busy now as I went through this in June.
Make sure all your paperwork is correct. I had an appointment to register the birth and then go to London to submit the passport app on the same day, and I found out on the day, they had to post the birth registration to me, meaning i would not be able to submit. Make sure you go to an office that issues the paperwork on the spot.
Also they don't advertise this, but they will not accept a counter signatory who is on mat leave, or on a career break, even if they are still an active member of the professional organization. Our first application was turned down on these grounds.
Sorry to bombard you with this, but getting a passport absolutely consumed the first 3 weeks of dd2's life. It was Hell. And you have a lot more to than I did, as we were already overseas, and I just went back to have the baby.
Also try and connect online with a baby group in your new home - try FB. Your new baby will need vaccinations a few weeks after arriving and you will want recommendations of dr's ect. The HV in the UK told me being a few weeks late wouldn't matter ( if you are nursing ) but don't leave it more than 3-4 months.
Finally - Ask for and accept all help! I have been there, take what you can, and good luck.