Does this hold up?

Data Used
The data we're using are the 40%/60% of double vaccinated and unvaccinated (strictly, less than 2 jabs) hospitalisations famously cited in the government coronavirus briefing by the Chief Scientific Adviser, Patrick Vallance.
Also needed are the rates of vaccination uptake for two doses, which as of today is 71.1%. See https://coronavirus.data.gov.uk/details/vaccinations
That's it. This is a very simple analysis.

Statistical methods used
We wish to find the probability of being hospitalised with covid given you have both jabs and compare it with the probability of being hospitalised with covid given you have less than both jabs (herein referred to as the unvaccinated population). What we have with the 40/60 statistic is the probability of being vaccinated/unvaccinated given that you are hospitalised with covid. To the untrained eye, these are the same thing. But they are absolutely not - the former considers the entire poplulation and the proportions of vaccinated/unvaccinated. The latter only considers the hospitalised population and therefore tells you nothing about the proportions of vaccinated/unvaccinated in the general population.
To find what we're looking for we can use the law of conditional probabilities or its reformulation, known as Baye's rule or Baye's theorem. As the latter is more intuitive for these purposes, we'll go with that. For more information, see its wikipedia page https://en.wikipedia.org/wiki/Bayes%27_theorem . Preempting "you cited wikipedia as a source", Baye's theorem is a foundational law of statistics, if I were to use it in a paper it would not need to be cited at all.