The UK sub building pattern is building their SSN class then their SSBN class then back again to maintain the ongoing capacity/ capability.
There are already points of significant convergence between UK subs and US subs plan going forward: my understanding is that the next classes of US and UK SSBNs will share the same (US designed and specified) missile sections while the UK appears to be aligning with future US reactor designs (presumably still to be built in the UK but very much the US designs).
In that context proposals to have a tri-nation future SSN is really a case of becoming a junior partner/ sub-contractor in producing the future US SSN which the UK would then effectively licence build in the UK. It makes sense from a US perspective (more subs produced for improved economies of scale, can cherry pick the minority content from the UK that suits them, etc.) and for the UK it would have significant cost advantages by greatly reducing duplication of effort/ overlap with the US etc. In fairness this may be seen by the UK as the only affordable and/or realistic direction to go in.
But such a move would inevitably eliminate the realistic ability of the UK to regenerate that independent capability and capacity (much more so than, say, even an aviation equivalent of ditching the Tempest project and the UK becoming a junior partner in the next generation US combat aircraft project).
Too much independent infrastructure, experience, knowledge etc, that couldn’t realistically be rebuilt would inevitably go away; the UK won’t be able to turn the clock back and will be dependent on the US for future submarine classes with the proportion UK content subject to future negotiations.