I don't think that the majority of people have actually had a change in their finances, so the only thing that is impacting on whether they can pay school fees is fear.
If people have actually lost their jobs and don't have prospects of securing a new one with the same level of income within a reasonable length of time, they are going to look at pulling their children out.
If people have suffered a pay cut, then they will work with the schools to keep their child in the system.
For people like DH & I, our salaries are not being cut, and we should still get small increments and have prospects for promotion. I was worried about monthly outgoings for a bit as it seemed that inflation was taking off, but that seems to have calmed down now. We may even be slightly better off because of the mortgage rate cuts.
Families who are committed to independent education will do their best to maintain the status quo. They will not make knee-jerk decisions, and school fees are usually the last thing to go. If you look at the trends over the last few recessions, you will see that pupil numbers did not fall during the recessions, and there were very slight dips afterwards, probably due to new pupils not entering the system as soon as they might.
If you are serious about doing proper research, I suggest you have a look at the Independent Schools' Council website - they have real data there, rather than third hand anecdotes.