If all the Indy schools remain full, then there isn't really a problem! What we're looking out for is fewer kids in private as a proportion of state - since that shows a change in parent school choice and will stop the policy from raising the money that Labour expect.
If the percentage of kids going private instead of state drops by 5% long term (ie 6.65% of kids go private instead of 7%) then the amount raised will be half what Labour said. At 10% (ie 6.3% go private instead of 7%) , the tax makes £0. Any more than that and the policy loses money.
This year might still not reflect the full long-term change, but I think that most people at a natural entry point apply for both state and private if they aren't 100% sure, and then see what offers they get. By the time the offers came out this year, it looked likely that Labour would get in, and they'd published their plans for VAT. So looking at YR and Y7 (and maybe Y12) should reflect what parents will choose if they aren't already tied in.
Of course, some people might not have applied to state/might not have tutored for grammar in time/might have accepted or emotionally committed to private before knowing about VAT, so it will still be an underestimate of how much movement there will be eventually. But it should give an indication.
At a rough guess, I'd say that if the proportion of children starting private in YR and Y7 next September drops by 5% compared to this year, that would be a red flag to stop the policy before real damage was done.
The loss of students won't be evenly spread, even locally: some schools are more popular than others, so if there's an overall reduction of 5% then some schools will still be full (with some kids getting in who wouldn't have in previous years) and less popular schools will be down 10-20% (assuming kids have 2-4 schools to choose from within reach). A school might survive a 10-20% drop for one year, but not long term.
I'd also look at a local level whether LEAs have managed to place all the students applying, to check for potential pinch points in the state system where there is a high percentage in private - like Surrey, Edinburgh and Bristol - which might require bulge classes or even new schools which would reduce the financial benefit.
I'd be more focused on standard entry points than on how many students currently at private school were moving at a non-standard entry point - since that's a short-term anomaly - but I'd still want to check other years too to see whether the state schools were coping locally.
I'd also aggregated nationally, to even out local differences such as specific problems with schools, catchment areas etc.
I'm not sure how it would be possible to check what type of students were migrating - eg whether the fears came true that students with SEN (who cost more to educate) would migrate in disproportionate numbers. Perhaps checking how the number of EHCPs applied for has changed compared to last year would give a rough idea.
That's where I'd start anyway!