There are three potential issues that make this ATL difficult.
1. The desire is for SSBNs. One could argue that if (for some reason) the RN decided not to adopt nuclear propulsion, they would rather have to build a larger number of smaller ballistic missile submarines with a smaller missile payload. However, the prequisite is for SSBNs.
2. Building 12 SSBNs is one thing, preserving them until 2021 would demand a whole different threat enviroment / geopolitical situation to justify keeping that many SSBNs in service. The Cold War never ended / limited aggression in Eastern Europe by Russia sometime during the past 20 years? Still, highly unlikely that the RN would be allowed to keep 12 SSBNs in service. The UK MoD budget would also need to be a lot higher to allow for that, meaning a different financial situation in the UK and priorities set by the government. I cannot imagine how the UK is supposed to pull that off...
3. Opting for a fleet of 12 SSBNs would mean for the British to eliminate all other nuclear strike options during the Cold War. Otherwise, it's an overkill.
No V-bombers after the 60s, no Thor IRBMs. I do not see the RN as being able to push that through.
One possible POD would be the early development of very accurate SLBMs, armed with conventional warheads and proven value in conflicts of the past decades. This is quite a difficult POD, since SLBM-launches could easily be mistaken for a first-strike nuclear attack attempt. However, if (by some POD) SLBMs were to become more popular and useful and replace cruise missiles, then 12 SSBNs armed with conventional warheads could become a viable power projection / strike platform. Keeping however 12 SSBNs for power projection in service to replace duties of RN aircraft carriers / destroyers / RAF Typhoons with Storm Shadows would require a whole different doctrine.
Problems with such an approach are certainly the higher costs of SLBMs in comparison to cruise missiles.
However, do look at recent developments in the field of accurate, conventional armed ballistic missiles (DF21-"carrier killer", Iskander platform).