I cannot pronounce myself on the likelihood of this in Britain (but we also shouldn't according to the OP), and I can only venture general speculations about the Soviet Union. I have quite a clear view on what this might have meant on the German political landscape, though.
I don't think anyone in the Soviet Union would feel very emboldened by this move. There was an officially tolerated, sponsored and orchestrated anti-rearming peace movement in less Moscow-loyal Eastern Bloc countries like Romania, and the way this went so much better than other official demonstrations clearly showed that secret services all across the Eastern Bloc that their populations were keen on peace and would readily jump on a peace train, too. The Soviet leadership would continue its military investments because Reagan's programs alone were enough of a perceived threat to the balance of mutually assured destruction, regardless of what the Brits did.
But EVEN IF they didn't, and that's a very big IF, the Soviet economy was in a dire situation in the 1980s anyway. They were losing ground technologically, their terms of trade beyond their bloc "partners" were deteriorating, raw resource prices of which the SU's territory was rich had fallen spectacularly, and the prices of loans on capitalist markets were rising. None of this would change just because less resources would have to be invested into military technology and could thus be directed to, say, consumer goods. Military technology was the only sector where Soviet R&D was competitive anyway...
Also, a more relaxed world political climate would only have benefited those who wanted a reformer like Gorbachev at the helm. With Gorbachev still coming to power and implementing his reforms, it's a given that the Eastern bloc satellites will try to break free and succeed, and then Russia is following suit.
I'm not 100 % certain with regards to the above. But where I am fairly certain is what this would have meant for the German political landscape. We had many hundreds of thousands, maybe even millions, demonstrating in Western Germany against rearming. This movement was closely connected and intertwined with other movements like the anti-nuclear power movement, the environemntalist movement in general, and movements for more direct democracy, feminism etc., many of which saw the Green Party as their parliamentary arm. If Labour had won in Britain on such a platform, and then went through with such measures, this would be an absolutely unprecedented triumph for the movement in the entire continent. It would boost their confidence greatly, and would have implications on all sorts of other political domains in which they were active. Quite generally, an end to Thatcherism in Britain and a switch to whatever kind of flavour of Labour would contribute significantly to an entirely differen narrative of what the 1980s politically stood for. IOTL, it was the decade in which neo-liberalism triumphed, in which the Western model unambiguously won over the Eastern model, and in which civic protest movements of all sorts were loud and visible and managed to insert their goals into public discourse, but failed to achieve any of them in immediate politics. With the change the OP alludes to, this narrative might change significantly, towards a narrative of a "third path" between American capitalism and Soviet communism becoming victorious: this would be a narrative with which both euro-communist groups and the electoral groups who supported Mitterand's leftist coalition and Britain's Labour Party on the one hand and civic protest movements closer to the Greens and certainly demonstrating against Mitterand's and any Labour PM's energy and infrastructure policies, on the other hand could both identify with. What with Chernobyl still happening and everything, it might mean Germany's Greens edging towards a double-digit electoral outcome in 1987, which might either have cost Kohl his majority for a second chancellorship (which would influence the course of German unification in 1990 considerably), or, if it is all to the SPD's detriment, would have shifted the weights within Western Germany's left-of-centre parties substantially. Foreign policy-wise, I don't think that this was a time in which a German government would waver in its Western orientation, even though the voices in favour of leaving the NATO (the protesters' demand was to "abolish it", but of course that's not for the FRG to decide...) would be slightly louder.