it's going to take longer to get through all the older age brackets somewhere where there is a large proportion of 50+ people
It's not about proportion. It's numbers.
A hamlet with 20 people in total and 10 of them elderly would have 50% elderly...
London has more elderly than anywhere else in the country. Lower take up rate though - which is why they might be doing younger in some boroughs (definitely not all).
Some parts of London have high levels of the Delta variant. I assume, if as a PP has says they're offering 16-18 in Bolton too, the government is trying to protect people in areas of particularly high risk.
Someone upthread suggested it's about protecting adults. Aside from the fact that it's psychologically protecting a child if their parents and other loved ones are protected (bereaved children suffer a lot), there is also Long Covid. Which does affect children too. A poster on another thread wrote yesterday about how it has affected her ds.
The EU has approved it for 12-15 year olds, Israel, UAE, and America are already vaccinating children.
The only issue is limited supplies.