To put some figures on the number of places, the most recent statistics showed 578,000 unfilled places in primary schools and 465,000 unfilled places in secondary schools. That might seem a lot. However, despite this, 17% of primary schools and 23% of secondary schools were operating at or above their official capacity. In total, these schools accommodated 59,000 more pupils than their capacity. The explanation for this is that many of the empty places are in the wrong location - a school with spare places in Luton isn't much use to someone living in Canterbury.
The ONS projects that another 30,000 primary places and 40,000 secondary places will be needed by 2027/28. This, of course, assumes that there is no change to the proportion of pupils attending independent schools. The most recent survey showed 592,000 pupils in independent schools.
So in theory, there are enough places in state schools to accommodate independent school pupils if they all transferred. However, in practice many of the places will be in the wrong location. Many more additional places would be needed.
Looking at the worst case for a Labour government, if all 592,000 pupils moved to state schools, the £1.7 billion they want to raise through VAT will not materialise and, in order to maintain current levels of spending per pupil, they will need to increase the budget for schools by £4.4 billion. This ignores the cost of any new buildings needed to accommodate additional pupils.
A similar policy was introduced in Greece in 2015. It led to schools closing and acute shortages of both teachers and school places in the state sector.