International Social Security agreements, often called "Totalization agreements," have two main purposes. First, they eliminate dual Social Security taxation, the situation that occurs when a worker from one country works in another country and is required to pay Social Security taxes to both countries on the same earnings.
The US and UK entered into an international social security agreement on January 1, 1985. I am pretty sure this mean that US citizens who lives in the UK does not need to pay US social security on the UK salary. The fact that you are a US permanent resident, and not a citizen, who lives in the US, may complicate matters. I am pretty sure you pay "Social Security" in only one country, but I am not sure if it would be to the US or UK. Your situation is sufficiently complicated, potentially evolving three countries (US, UK, and the country you are a citizen of), that you should check with a tax specialist.