I spent some time going through internet explorer settings and resetting things, and that has cured it. The strange (to my mind) thing is that it was only BT that it wouldn't connect to.
Is it trying to tell me something?
Previously I was with Orange. They are now EE for 'Everything Everywhere' and should be 'NA' for 'Nothing Anywhere'. A totally useless bunch of idiots, and in the case of their Indian 'help' centre, stupid liars.