A “back of a fag packet” calculation perhaps isn’t the best way to be looking at the impact on state schools. To be fair though, it’s probably the same calculation that the Labour Party have used.
Your figures are extremely skewed and have simply divided all private pupils between the entire number of schools, the vast majority of which are primary, some of which will be very small schools. There will be a much greater impact on secondary schools and sixth form education, because there are far fewer state secondary schools and a higher percentage of pupils attending private secondary school. The IFS estimate that 18% of pupils are privately educated at sixth form level for example.
There are currently around 30,000 state schools in the UK. However, around 3,000 are nurseries or early-learning centres. 21,000 are primary schools, and 4,000 are secondary schools. There are around 1,850 special schools and pupil referral units. These are rounded figures only.
So, any pupils moving into the state system at secondary level will be split between approximately 4,000 schools.
There are around 3.2m pupils in state secondaries. Assuming that the estimated number of pupils is private secondary is correct, at about 8%, that’s around 280k pupils. Assuming a quarter move to the state system, that’s 70,000 students across 4,000 schools. That’s just less than 20 per school. However, it’s not as straightforward as that. Lots of schools will have far more than an extra 20 pupils wanting places. The type of parents who are prepared to pay thousands each year for education are not the type of parents who will sit back and just happily send their children to failing schools and those with spare capacity. They will fight for the best state schools, by paying for tuition and high house prices in good catchment areas. All that will happen is that the more deprived children will get pushed out of those schools.
Aside from that, the state system can’t cope with the number of children they have now. An increased number of pupils won’t help that.