BBC and ITV continued with 405 past any reasonable date for switching them off. Switched off at start of 1985, duplication of services on 625 started in 1969, It would have been cheaper to switch them off some years earlier (perhaps 1980 or so) and give anyone legitimately still watching on 405 a new set.
I don't know the politics of the timing of the 405 switch off. "Granny watching on 405" was surely urban myth. By comparison the digital switchover was done from 2008 to 2013 though DTV had been transmitted as early as 1998. Set top boxes were given free to certain vulnerable and elderly people.