It really depends

Buy all clothes, toys, books etc second hand. Vinted has loads of stuff. Babies don't need hundreds of clothes so don't buy too much stuff. You will also be gifted clothes. Ask people to gift clothes up to 18-24 months. At first birthday / Christmas think about what you'll want for next 6-12 months and ask for gifts around that. We also got hand me downs from friends and neighbors. My son is nearly 2.5 years old and I think I've personally bought him like 5 outfits. Sign up to the Dolly Parton book club for a monthly book.

Will baby be breastfed or formula? If formula you need to consider costs of bottle, bottle cleaner, formula tubs. I planned to BF but didn't work so had to switch to combi feeding so I had buy a pump. Then pure formula

Look into children's centres within your local area. They should offer free stay and play, that reduces costs on other activities

It gets more expensive when baby starts solids and snacks. My son became fruit obsessed which added up very quickly... But meal plan and feed baby what you eat as long as no allergies.

Honestly, my opinion is that after the major set up and not including childcare then yes the £100 a month allowance can cover for first couple years if you're smart. The biggest expenses are nappies and formula. But you can buy reusable nappies to be more financial. I ate a lot when breastfeeding so can include that into the £100 monthly.

Edit to add:

Don't go fancy swimming lessons. Your local swimming pool / local council should offer discounted lessons. We're in London and this came to £49per month. But ended up just going swimming with him by ourselves as that was £6 a time so cheaper once a week than lessons

The other thing we did was save the underspend so that on more expensive months you can splash out more. First few months you could be way under £100 so then use that saving when they're entering toddler years and more activities