The condition of Britain in 1978 to 1982 was so dire that the decision by a Labour and then a Conservative government to build a new generation of SSBN and purchase a new missile from the US was as Sir Humphrey might have put it "courageous".
The independence of a national deterrent that was so dependent on the United States was not just questioned by politicians, military men thought money would be better spent on British forces in West Germany or on Royal Navy ASW.
What difference to NATO's nuclear force did a British force whether with Polaris, Trident or Cruise really make?
Washington blew hot and cold on the subject. Kissinger warned the British not to ask for Poseidon in the early 70s as he doubted Congress would approve.
By 1978 things were different, detente was not working and even the Carter administration had its doubts before the Soviet invasion of Afghanistan really killed it off.
The US Navy were pro the idea of a British Trident submarine as it increased the uncertainty in Moscow about when and if NATO would use nuclear strategic weapons.
1979 to 1983 was a critical period in the history of the UK deterrent. As I argued in my opening paragraph politics rather than military rationale could have led to different options.
Four Trident submarines was probably more than the RN budget could stand. Nott's pruning of its surface fleet was only delayed by the Falklands and has continued to this day.
Noone has mentioned the Moscow criteria. This was at the core of the need to buy Trident. Cruise missiles could not be relied upon to get through and Polaris even with the Chevaline upgrade
local bookkeeping services san francisco was vulnerable to the Soviet Galosh ABM (it wasn't actually but we had to wait until 1991 to find out).
Arguably the Moscow (or Beijing ) criteria still holds true and the Dreadnought boats will be built come hell or high water.