My vaccination booked for tomorrow is cancelled - just got the email saying the vaccination centre will be closed. So I've rebooked for early October. Might have got a new appointment in late September if I'd shopped around for a different vaccination centre, but less convenient for public transport and no guarantee that will be open on that day anyway.
Have you checked around local GP clinics & pharmacies in your area. Many are now providing vaccinations and often have times available earlier.
Below are appointments currently showing as available for the week commencing 5 Sept on the H365 website for our local GP clinic.
I think I'd be shopping around to see where I could get it sooner than later.