It was to do with that. There was a fascinating documentary about it a while ago. Macmillan was desperate to get access to US technology to ensure we retained a seat on the security council as we'd fucked up developing our own delivery mechanism and the Americans abandoned their own sky-launched stand-off. They negotiated the Nassau Agreement to take Polaris but the Americans extracted a high price. The main prize was the Holy Loch base, which scared the shit out of the UK government as it was so close to Glasgow and the Clyde but they had no choice.
The other major concession was that it was part of a multi-lateral force (i.e. effectively under the US umbrella) as there was no way Kennedy and McNamara would countenance a truly independent force outside of US control. But there was a vague form of words that we could operate independently only when our "supreme national interest" was threatened. But no one really defined that properly and to all intents and purposes it translates as "when the missiles have already landed and we're completely on our own because we can't get hold of the Yanks to say yes or no". The Nassau Agreement has not, as far as I'm aware, ever been superseded.