My thoughts, for what they're worth ... you're talking about physio for injuries caused by running or other sports, yes?
I've always gone private. I'm fortunate in that I can afford to, but it has never occurred to me to go to the NHS with a running injury (even though my GP is a runner and at my very rare appointments we spend more time talking about local half marathons than whatever is ailing me).
I've always assumed that NHS physios are for much more serious matters than eg a dicky hamstring - regaining mobility after a car smash, cancer operation etc. I've also always assumed there is a long waiting list.
The local sports clinic, just down the road from me, can usually do an appointment the same evening.
In an ideal world the NHS would have resources to cater to the active and fit as well as treating the lardy burger-munching chain-smoking couch potatoes for the consequences of their own poor lifestyle decisions (I generalize, but you get the drift). But it doesn't.
So I suspect you might have difficulty comparing the two but I wish you luck with your project.