I've already sent one Nest e UK thermostat back to the seller, due to it not turning on. I was told to leave it on overnight but it had no sign of charging or turning on, even by morning.
I received a new one today and I am experiencing the same issue. It is plugged in but appears to be dead as a dodo.
Just tried double pushing the display and the Google logo came on and the the light flashed on top. After a few seconds it went back to the charging icon and the light stopped flashing on top. After a few seconds more the screen went of again and it appears to be dead, until I single press the display and the charging symbol comes on again. Will leave it for now.