You'll still have the window where efficacy takes 2-3 weeks to kick in, so rapid vaccination won't contain an outbreak of cases but it'll definitely have some impact. I think generally a speed up of the first jabs country wide needs to happen, as we'll see more cases spread but the offset of that is that the most vulnerable are now protected so it's difficult to see the current approach from PHE changing dramatically.
I think this is a test case for allowing local fast response to problems. The case numbers jumped rapidly in certain postcodes, going from around 20/day across the whole of Bolton on the 20th April to 40/day by the beginning of May, to over 100 day by the 9th - almost all concentrated in certain postcodes. The local health authority reacted at the start of May, but not with vaccines for the under 40s. The cases so far have been mostly in the under 40s.
Combine that with low vaccine uptake in those same postcodes, the ability to quickly allocate supplies to the area could have given us a way to slow the growth with basic protection starting to kick in around now. Whole families getting vaccines at the same time might help break the poor takeup issue etc, a no-booking system using local facilities might help. That's being wise after the event in Bolton's case, but that's no reason not to start now - in those postcodes and beyond, particularly those where takeup so far has been weak. Local response should have more tools than just putting up signs asking people to wear masks in shops and some more LFTs.
It's just irritating hearing the clamour to put Bolton (or indeed the whole of Manchester) back into lockdown at the same time as the local health services are waiting for permission to act and Boris is announcing hugs and no masks in schools.