//Personally I'd say Yes,it would be pointless opening everywhere up when the time comes if infection is on the rise in places like London//
Infection rates are not increasing in London:
https://www.bbc.co.uk/news/uk-england-london-54324244
You will see that only two of the 32 boroughs showed an increase. These were Islington and Tower Hamlets. Both increases were small (2 or 3 cases per 100k) meaning the increase was probably statistical "noise" and those two boroughs are near the bottom of the list of London boroughs anyway.
//I am suggesting much stricter boundary control.//
Absolutely impossible in large parts of the UK. I live almost on the border of my Local Authority. My nearest large shops (just over a mile away) are in the adjacent LA. The nearest shops in my own LA area are three times that distance. My nearest hospital (and where many of my neighbours have been called for their Covid jabs) is in the adjacent LA, a twenty minute walk. My "own" hospital is almost five miles away, two buses for those with no wheels. We don't do county boundaries.