IIRC this was done by threatening to leave NATO, which put pressure on the USA who really didn't want to lose a perfect base location to help monitor/contain the USSR, who in turn put pressure on the UK.
But this is from memory; and given the sources I read may have been politicised (I'm British by birth), take with a pinch of salt.
Nah, that wasn't it. It was done by Iceland having shorter supply lines and both sides not willing to start killing each other. https://en.wikipedia.org/wiki/Cod_Wars
