You need at least one more layer of math there. You need to account for percentage double vaccinated vs unvaccinated.
Because of how there is a lag between being double vaccinated in the data of 14 days, and the UK's vaccination rate, there isn't that big of a difference. But if you want, let's look at the last two reporting weeks. Between June 7th and June 21st you have new hospitalizations of;
Unvaccinated 1,182 - 251 = 931
Double Vaccinated 313 - 42 = 271
Unvaccinated Rate is simple from June 7th to June 21st
Unvaccinated 39.1% to 35.3% = 37.2%
Double Vaccinated Data is >= 14 days from second dose, so it's actually May 24th to June 7th data on vaccinations in arms.
Double vaccinated 34.9% to 42.4% = 38.7%
Unvaccinated 931 * (38.7%/37.2%) = 969
Vaccinated 271 * (37.2%/38.7%) = 260
(969 - 260) / (969) = 73.2%
Happy?
Overnight hospitalization rate during these two weeks is nearly double for double vaccinated people as compared to unvaccinated.
Unvaccinated 931 / 251 = 371% * (38.7%/37.2%) = 386%
Double Vaccinated 271 / 42 = 645% * (37.2%/38.7%) = 620%
Likely do to a combination of things including letting their guard down now that they are "safe", and the overall increased susceptibility of the age groups that have been double vaccinated.
Also you should look at the health/succeptability differences in the two groups (vaccinated has a lot of 94 year old overweight diabetics) but that is much harder to quantify.
No government has been willing to split the risk factors and measures based on age groups. Which is why this time last year we already knew the average age of death was over 80 world wide, average hospitalized age wasn't too far behind that either. But we still applied blanket measures on the entire population.
Good luck with that. It doesn't support his narrative.
It actually didn't make a statistical difference, so I kept the math simple.