This was in the 1950s and is described in Peter Hennessey's excellent book The Human Button.
It's been a while since I read the book, but from memory the cabinet office also purchased an AA membership so that the prime minister's driver would have a key giving them access to the private AA phone boxes that were fairly common back then. They had an arrangement with the AA call handlers to allow the PM to be connected to the appropriate part of government in an emergency.
Meanwhile the US and Russian government probably had satellite...