Another sleeper story in the English locals results is Con decline and LD/Lab advance in "blue wall" constituencies which voted Remain in 2016 but returned Con MPs in 2019.

Here are changes in vote share, broken out by which party is in second place in Westminster:
Con Remain seats where LDs are second:
Con +2, LD +4 (seats last fought in 2016)
Con -7 LD +3 (seats last fought in 2017)

Con Remain seats where Labour are second:
Con +2, Lab +4 (seats last fought in 2016)
Con -4.5, Lab +6 (seats last fought in 2017)

(those are average changes across wards in the relevant category)