November 2021: Are current UK COVID case numbers inversely correlated with vaccination uptake on a regional level?

An odd trend in the latest available gov.uk data on COVID 7-day case rates and double-dose vaccination uptake by lower-tier local authority (district/borough councils and UAs).

There seems to be no positive correlation between a low current case rate and a high vaccination uptake. Indeed, a mild negative correlation unexpectedly presents, and the areas of low infection rates and low vaccine uptake are dominated by London, Manchester and Birmingham - densely populated areas where the virus would be more likely to spread quickly. Is this herd immunity from infections?

Source data: ONS and gov.uk Coronavirus data dashboard - data accurate as of 24 Nov 2021.