If "death and total counts increased for both the vaccinated and unvaccinated groups" are increasing as you say, it still fails to account for the difference in the ONS vaccination rate and the UKHSA vaccination rate - which is, on average half.

I might ask the ONS, of course. In the meantime, I am going to put my workings out to a wider audience. I hope you don't mind if I link to our discussion: it was quite fruitful, I think, and someone more expert might be able to shed some light.