Hurrah for your new arrival!
My partner and I have been tackling the coast to coast in sections since our baby was 3 months. The bits of advice I'd suggest are:
1. You need a really good carrier. For me, that absolutely didn't mean one of those baby bjorns, or a rucksack style carrier. Because both of those carry the baby with their legs dangling, they were really hard work after the first few miles. I wanted one that carried the baby with their legs wrapped round my waist. There are loads on the market, and they are much cheaper than rucksack or bjorns too. I found a Mei Tai most comfy, but my partner preferred a Soft Structured Carrier (these are types, not brand names) as it had buckles and looked more 'proper'. They can carry everything from newborns to 4 year olds, in the same carrier. I got mine from www.mama-licious.co.uk And it's really imortant to try the carrier on before you buy it - different carriers suit different people.
2. Make sure you can feed your baby on the go. If your baby is small, they'll need feeding lots. Whether you are breast or formula feeding, you need to make sure you can feed whilst walking, or you'll do a lot of sitting perched on a rock with a numb bum, and then a little walking. So make sure you can feed in your carrier. If your little one is older, a toddler, make sure you carry lots of snacks to keep them entertained. I found rice cakes and mini-packets of raisins were lightweight, interesting, nutritious, and survived being dropped in puddles ;-)
3. Forget cots. Some places offer them, most don't. We considered buying really expensive pop-up travel cots, but in the end just found having the baby in bed with us was a much easier option all round, and meant he settled in strange places much better.
4. Suncream and hats. If you are carrying your baby, they'll generally stay pretty warm, so I'd worry more about sun than other weather. If you take just one sun hat you will lose it. Probably within 5 minutes of setting off. If you take 3 it'll be cold all through your walk and you'll never need even one (life is just like that), but at least your little one won't get sunburn
5. Folding changing mats are a great idea. There's few places to change a baby en route, and even where they exist they're rarely open at the times your little one has poo-ed. Changing on the ground is normally fine, but not when it's just rained...
6. Consider ways to make they journey easier. Walking with a baby can be great fun, but it can be tiring too - you have to go at someone else's pace to some extent, and you are carrying a few extra pounds in baby, clothes and nappies. Plus you will probably still have to get up for night feeds too if you baby is little. So work out ways to make the things surrounding the walk easier. This is one time when I think using a sherpa can really be fully justified, or not doing self-catering if you normally would, or (for us), doing the walk in shorter stages, taking maybe a weekend or long weekend to do just one stage of the journey, and coming back to it again and again.
Enjoy it - there's noting like the joy of sharing a walking passion with your whole family, and having a child really doesn't mean you can't continue to enjoy some serious walks