There is quite a lot of law regarding the tracing of birth parents and offspring and it can get quite involved. I would suggest as a starting off point you cantact the adoption worker at your local Social Services Department. They are expert in such matters and will handle things very sensitively.
I once ran a class on family history and one of the attendees, who had been adopted and traced his mother, was going to drive to Scotland the very next day, knock on a door and say "Hello, I'm the son you gave up for adoption." In the cicumstances, not only would his actions have been devasting to the old lady, they would also have been illegal.
It is not beyond the bounds of possibility that your grandmother is still alive so I would make it a priority to contact social services. If you do manage to find her do not try to make contact yourself, leave it to the experts. Also Social Services can access records that ordinary members of the public will never see.
If you are unsure, PM me and I will send a short piece about what the law requires; if I can find it in my notes.