I moved when ds was 5 months, not a lot of fun but survivable.
Re: addresses, arrange a mail redirection for a few months at least, gives you some slack to sort out notifying everyone. This includes CTF, I don't think I've got round to notifying them yet (and we moved 6 months ago).
Register with a GP when you arrive - I had to do this literally on day one because dh is chronically ill. They SHOULD notify the hvs but it won't do any harm to ask for the info about the service and then just present yourself at the next convenient baby clinic with your red book (and baby, obviously).
Don't try and do too much, it is knackering trying to unpack with a baby to look after. So make sure you pack up smartly, boxes with essentials only, and let things take their course. I have boxes which I still haven't unpacked. Clearly not that important!
I assume you're not moving yourself - getting the removal firm to do the packing is an absolute godsend. I remember having to organise it round rooms so that ds had a place to nap whilst it was going on, all very complicated.
Are you moving from owning-to-owning, renting-to-renting or some combination of the two? Make sure you have meter readings for both properties regardless, you'll also need to notify the Council Tax people (although given the speed with which they've dealt with the tenant in our old property, not with any particular hurry!) and I'd probably prioritise car insurance, as the new postcode could affect your premium. Make sure you have contents insurance on the new place too.
Good luck!