There was a roundtable discussion on BBC's Newsnight after the airing of Threads, you can watch it on youtube here: https://www.youtube.com/watch?v=Y0VcT-XWb7M

I would say that this debate adequately encompasses the diversity of opinions on the topic among the politically active and among folks in government (which were represented in the panel).

There are basically three different points of contention.

Perhaps the biggest one was the theory of "nuclear winter". This is the theory that the dust kicked up by nuclear detonations and injected into the stratosphere combined with the smoke created from burning cities would linger in the atmosphere long enough and cause sufficient dimming of sunlight globally to affect the seasons, causing mass die off of plants, and endangering food production (among other things). This theory is still unconfirmed today, and in the '80s as you can imagine it was common for folks who were opposed to the use of nuclear weapons to believe in the theory while "pro-nuke" folks tended to disbelieve it. As this was a major plot point in Threads, contention related to the proof or disproof of the theory affected the degree to which people took the movie seriously.

Secondly there is the performance of the "civil defense" apparatus in Britain specifically, and its ability to hold the country together after a devastating nuclear attack. Here the movie definitely portrays a bleak reality, where Britain's civil infrastructure falls apart fairly rapidly, and where a breakdown of order leads to chaos, looting, violence, etc. There was a lot of debate in Britain on a variety of related issues. The extent that Britain could possibly and should prepare for a massive nuclear attack (building public shelters, preparing emergency plans, food supplies, etc.) The reasonable likelihood of "success" of such plans in the face of an attack, even one isolated on Britain alone. And whether it made sense to invest heavily in such programs. At the time Threads came out Britain did not have an extensive shelter system, and all but a few people agreed that in the event of a large scale nuclear attack Britain would be in dire straights indeed, regardless of the effectiveness of any planning that had happened beforehand. Just as prior to WWII, in the 1980s Britain was still significantly dependent on importation of food to support itself. Any reasonable assessment of Britain's fate after a large nuclear exchange would involve a significant reduction in population. There was a tiny but vocal minority who felt that with enough preparation and planning Britain's chances could be elevated significantly. And at the time those people were closer to those in power, but no serious efforts came of it.

By far the largest point of contention rested on the nature of geopolitics at the time and the degree to which conventional conflicts could escalate to nuclear conflicts and, perhaps even more importantly, whether nuclear conflicts involving major Cold War powers (e.g. NATO or Warsaw Pact members) would inevitably escalate into full-scale "MAD" nuclear exchanges. The conventional wisdom was that almost any direct conflict between the US and USSR specifically would almost certainly escalate into the use of nuclear weapons, and any use of nuclear weapons involving NATO or Warsaw Pact members (specifically involving each other) would escalate into a full-scale global nuclear war. There were, however, many people who felt that there was a potential for more limited conflicts, even including use of nuclear weapons, to occur. Whether this sort of thinking makes sense given a backdrop of "MAD" and deterrence threats as the operating principles of the day aside, they were reasonably widespread, and the ideas were close to the governments in the US and the UK at the time.

Of course, there was also a considerable amount of interplay between this point and the point on nuclear winter, since if even a "limited" nuclear exchange could doom the world then whether or not it escalated into a larger scale exchange was somewhat academic.

So there were people, even some in positions of power, who felt that a nuclear war might be more survivable than a movie such as Threads would portray because a large scale nuclear attack was not an inevitability. Even more so, there was a much wider-spread thought that a massive conventional conflict between the US and USSR or NATO/Warsaw Pact (say, threatening the East/West "front-line" in Europe) was possible without it necessarily escalating into a nuclear war. Though, fortunately, such theories were never put to the test during the few remaining years of the Cold War.

As for Thatcher, I'm not aware of any direct comments about Threads specifically (though they may exist), but she has been on record as believing in the utility of nuclear weapons, as in this quote: "A world without nuclear weapons would be less stable and more dangerous for all of us." Additionally, in 1981 (during the Thatcher government) the UK acquired Trident missiles from the US which it used to upgrade its submarine based nuclear missile forces. For context, this was a significant upgrade of the UK's submarine missile forces as the Polaris missile could carry only two warheads while the Trident had greater range, greater accuracy, and could carry up to 8 warheads, though ironically they didn't enter service in the UK until after the Cold War was effectively over.

There was, as you say, a pretty strong decline in approval of UK's nuclear weapons arsenal from the mid-1980s onward, perhaps attributable to Threads specifically, perhaps to the general sentiment at the time. Nevertheless, the Conservative government did stay in power after the 1987 general election, so there was certainly a limit to whatever anti-nuclear sentiment existed in Britain at the time.