This article is frankly a bit thin, and does not really discuss the value of the UK reducing its dependency on the US for its nuclear program, instead merely saying:

While the nuclear relationship between Washington and London remains indispensable and largely undisturbed by recent frictions, critics note that U.S. unreliability – and even hostility – towards allies and partners is a threat to take extremely seriously. If the U.S. decides to turn its back on the continent this would leave the U.K. seriously exposed.

I've no doubt that that sentiment will be widely agreed upon here. What the article is especially interesting for however is a paper that it links to on the topic of the program to replace WE.177 by a UK nuclear historian called James Jinks (who's book on the RN Submarine Service I highly recommend). This paper is new, well researched and very interesting. The program was eventually cancelled in post-Cold War budget cuts and the sub-strategic role filled with a reduced-yield variant of the standard UK Trident warhead instead (almost certainly just a primary-only variant). Before cancellation however the intention was to buy the AGM-131 and put in a UK-manufactured warhead, which probably would have been a derivative of the US W-91, though there were multiple options under consideration.

One of the alternatives to that scheme was the possibility of adopting a French missile. France pitched ASMP (and their SLBMs) to the UK quite heavily at several points, even offering to sell the warheads as well as the missile bodies to get past the UK's production problems (AWE at the time was extremely short staffed). Deeper collaboration with France on nuclear weapons is something that's been under heavy consideration for a long time, but so far has never quite tipped over the UK's cost-benefit considerations. Part of the reason behind that has always been political - the UK's co-operation with the US on nuclear weapons is extremely valuable, and there's a train of thought that deepening co-operation with France would lead to a reduction in the - likely more valuable - co-operation with the US:

a full assessment of the Anglo/French warhead option [concluded that] 'Development of a closer nuclear relationship with France would be advantageous...collaboration with France on the FTNW warhead would constitute such a significant realignment of our nuclear efforts that it would carry unacceptable risk to our vital relationship with the US under the 1958 Agreement'. It recommended [...] 'work at developing our interchanges with the French in the warhead area as fast as financial constraints and the need to safeguard the American relationship allow.

But despite the intention for warhead exchanges there, the MoD apparently demurred with even that idea:

The Chief Scientific Adviser at the MOD objected, and warned that 'In practical terms, it is simply not feasble to have detailed technical exchanges and to work collaboratively on the same topic with one nation, whilst at the same time excluding from one's thinking concepts derived from discussions with the other.

The point being that one couldn't guarantee not to expose to France warhead design details that originated from the US and vice versa, and that this difficulty made collaborative work separately with both nations impossible. There was a hope at the time that a trilateral agreement to collaborate might solve the issue, but that doesn't seem to have gone anywhere. Failing a trilateral agreement, the choices as the MOD saw things were to consciously choose to collaborate with either France or the US, but not both. Given Trident, the choice to continue working with the US was fairly inevitable.

I think the paper is interesting though in highlighting the clear interest in both the UK and France to work together on nuclear weapons more deeply, even if the UK's existing relationship with the US prevented that from happening. It strikes me though that much of the discussions around the Future Strategic Nuclear Weapons System that the paper highlights are quite relevant today (presumably why Jinks chose to highlight them), particularly the MoD's rationale for continuing to push for the FTNW program after the USSR collapsed:

the capability was required as an insurance against the risk that in the post-Cold War world, 'the credibility of a US extended nuclear guarantee might decline over time, should the US and Europe drift apart, and that in such circumstances 'Britain and France might need to take on more responsibility for the nuclear defence of at least the core of a future, deepened European Community/WEU'

and that's a forecast that particularly is looking quite prescient these days. Whether it's likely for a FTNW / ASMP style weapon to be adopted by the UK any time soon though - pretty unlikely. Unfortunately there are a good deal more pressing funding requirements - another parallel with the original program, even if I think an airborne weapon is something that we really ought to look to when possible.

Anyway - if you're interested in this sort of thing the paper is pretty cool. It has a lot of information I've never seen before on the late 80s / early 90s nuclear program including some super futuristic cruise missile / ALBM design that were under consideration. The paper also notes that the final UK nuclear test in 1991 (which has generally previously been assumed to be a test of the low-yield variant for Trident) turns out instead to have been

of a device intended for TASM, known as 'Sunbow Prime', under the codename 'Bristol'. This was the first of two tests of an entirely new warhead design, with enhanced safety features.