Your estimates seem on the low side.
£3100 is probably doable for renting. I stuck a few SW zone 3ish neighbourhoods in and you can find 3 beds for that.
£400 for a food shop for a family of 4 is very low. Groceries are expensive now. I'd estimate around £150-200 a week minimum even if you're careful.
Contents insurance.
Car insurance is v expensive these days. It's gone crazy. You definitely need to allow for that.
Swimming lessons seem on the low side, I'd budget about £10 for a 30 min lesson per child.
Your after-school estimates are way too low. I'd estimate about £15 per day per child, so for your two for 3 days a week that would be £90 a week, not £70 a month! Or getting on for £400 a month.
Yes schools will probably teach French.
Adult haircut - a cheap price is £60. You can pay as much as you want of course. Halve it for kids. If you want a cut and colour I'd say you're looking at at least £120 for anywhere decent and possibly £200-300 if you want something that isn't a simple all-over dye (highlights, balayage or whatever).
You'll probably want to allow transport costs other than your commute. Getting around to activities etc at weekends if you take the bus, train etc. Not sure why you're not including petrol in this overall.
I would budget at least £220 per month for energy on average over the year. Don't think you'll get away with low bills in summer as it's just so expensive now. I pay £85 every month in a one bed one bath!
I'd allow £100 per month for two adults to get lunch at the office twice a week. And £20 per person for lunch out on a family outing.
Others have provided good lists of things like clothing you'll need to budget for. That will probably include school uniform and some schools also require specific uniform gym kit too, so budget for that each year as the children grow.