The UK and France have nukes; but (1) not many compared to the rest of the planet having a war with them, and (2) MAD still prevents industrialised production of super-greenhouse gasses. Russia has nukes but I don't think they're helped by AMOC anyway and thus wouldn't care.

The scenario of "post AMOC collapse Europe tries to compensate with more global climate change" is an existential threat to at least some other nations, how many depends on the scale of the change caused by the AMOC collapse and thus the scale of the change from the super-greenhouse-gasses produced in this scenario to compensate for it; even if the UK/France nuclear arsenals were significantly larger, this would not be sufficient to deter India from using nukes in this scenario if conventional methods failed.

Also, again, deliberately trying to push through an AMOC with even more greenhouse gasses is something most people don't model, so all parties would have to seriously consider if this would be at the level to melt global ice-sheets causing a (58m Antarctica + 7m Greenland =) 65m sea level rise rather than the much smaller one we'd expect in a business-as-usual scenario.

Set this map to 65m: https://www.floodmap.net

Then consider each scenario. If the AMOC fails, it's an open question how severe that even is — I've seen estimates between 2°C and 30°C of local cooling, and I expect the lower range of that hence my previous statement "Europe north of about Paris will look like southern Canada already does".

At lower levels of that range, Europe just goes back to where we were a century ago, at higher levels it could only be "solved" by greenhouse gas emissions that would cause the Antarctic and Greenland ice sheets to fully melt. Looking at the global map, my guess is that Spain and France end up like the USA Pacific coast, which isn't too bad, while the UK and Ireland have severe issues.

But even a "mere" extra 2°C globally is enough to be an existential threat to certain populations including in parts of India, who would do whatever it takes to prevent that.

If it's really a 30°C collapse — and it would have to be on that scale to seriously consider global geo-engineering rather than retrofitting buildings and building greenhouses over the farmland — then that risks the aforementioned ice sheet collapse, which neither the UK nor Ireland can risk because that would turn the problem from "it's cold and we need to import food" to "and now 50% of the country is underwater", likewise half of Belgium, all of the Netherlands, much of Denmark, and a significant part of the most densely populated regions across the continent including Italy.

France, the only other nuclear armed state in the EU besides the UK, is only affected a little bit by the sea level rise, and thus the only one which might consider that course of action. But even they would have a serious engineering project to stop the Seine from flooding back to Paris. And even with that, silver bullets (your nukes) can't stop golden guns (other people's nukes).

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No, if this particular disaster happened we might try to mass-migrate to Africa (to a chorus of "oh no not again" from the destination), but I don't see our governments trying to solve it in the "old lady who swallowed a fly" approach.