My DC went private for primary and then to grammar for secondary. Ideally I would have sent them private all the way through but I can't afford it since getting divorced (exH won't help). Therefore I'm fully in favour of private schools but given the huge cost I may have made a different decision at age 4.
A few thoughts for you to consider:
Have you calculated whether you can afford it through to at least age 11? The school's website will usually publish fees for each year group and there is usually a jump for y3&4 and sometimes again for y5&6. So if reception fees are c£12k a year (so c£4k a term) then y3 plus may well be c£15k a year. Then you have to factor in the increase in school fees. In the 6 years my DC were at their last school fees increased by 25% (and that was before the inflation rates as they are now). So if the y6 annual fees now are £15k a year you need to assume they will be at least £20k a year by the time you get there.
If you're in your mid-20s are you likely to get promotions etc? Ie if your salary is likely to go up on a regular basis then the above fees increases can be better accommodated whereas if you're in the sort of job where it doesn't really increase that's more of a problem.
Whatever agreement you have with your exP on fees - make sure that it is in writing, signed by both of you and that it specifies a percentage of the fees and not just a specific amount (otherwise you will end up paying loads more over time). Think also about things such as - if that school turns out not to be right for your DC and you want to move to a different school, what are his thoughts on that. Try and factor whatever you can into an agreement now so it is harder for him to wriggle out of it down the line but equally make sure you have some flexibility if you change your mind. While it may all be amicable now in 5 years time it may be very different.
Is the private school local to you? What is its usual catchment area? Where do most pupils live? One of the difficult things for my DC was not living close by to their friends so play dates always took more organisation and were more difficult.
If you have a reasonable length school run have you calculated the cost of that? Not only the actual fuel cost but wear and tear on the car, time taken etc. I didn't calculate this and the school run although only c20mins each way worked out at c10k miles a year which with fuel costs as they are is expensive.
Is the private school one where there is a mix of parents and plenty of working professional parents (ie not just the super rich)? I assessed this by checking out the cars at school pick up time 😂. If it's mixed your DC will likely fit in more. It was at my DC school and the super rich were definitely the exception.
Have you factored in the much longer school holidays and childcare for those? As they are usually off for a few weeks in the summer before the state schools break up it can be v difficult to find holiday clubs etc and those that are available are expensive (as they're for prep school DC), Christmas and Easter are usually a week longer as well. The bonus to that is you can go on cheaper holidays (if you can afford any...)
The wrap around care in term time is usually a lot better - my dc could go 7.30-6 if necessary and have all meals at school and do their after school activities there so we didn't have to take time out of weekends etc to do that.
For us it's meant missing out on expensive holidays, replacing the car, doing work on the house - those sort of capital expenditures as I haven't been able to save additional money.
Personally I felt it was worth it and my DC were happy at their schools but if I had my time again I would definitely look more seriously at waiting until y3 (especially if you're in an area where they have separate infants/juniors schools and can find a nice infant school for the first 3 years).
Good luck with your decision.