It's a tricky one, with the Oxford vaccine they seem to think that the 9-12 week mark is the "sweet spot", and with Pfizer they haven't done those kind of tests as yet.
I don't have the article handy but I did read something about how they implemented this as a 2-shot vaccine because historically an initial dose with a booster has been shown to have the best results, and in a lot of cases (i.e. other vaccines) the booster is best taken several months after the initial dose - however doing the clinical trials along these lines would have considerably lengthened the time it would have taken to complete the trials so they opted for the shorter interval.
Bearing this in mind, there seems to be a strong suspicion that a longer period between the initial dose and the booster is maybe the better approach to do it, however because they didn't do the trials this way then they can't definitively say this (for the Pfizer vaccine, at least). The biggest factor in specifying 3 weeks was that it is the longest amount of time they wanted to wait between doses, not necessarily because they thought it was the most optimum time for a booster (as the Oxford vaccine has supposedly shown)
From my personal point of view, I would rather get more people a single dose initially and deliver the 2nd doses after 10-12 weeks - I think this gives a larger proportion of the more vulnerable some degree of cover, and that will help greatly.
Also, on the subject of the government doing this for political reasons so they can say we have vaccinated more people than other countries... I wouldn't put anything past this government, but in this instance I don't believe that makes a difference as it would be the same number of vaccines in the same time frame, regardless of whether it would be 1st or 2nd doses. I agree they are able to say that more people have had some kind of dosage, but I think it's splitting hairs at that point