I understand that it is possible to tell where a nuclear weapon was produced from the isotopes in the fallout. So the only way this would work is if the agent detonated one of the nukes on the base already. Arguably this would be simpler than smuggling a weapon into the country, but still a tall order.
I would say that Britain pulls out of NATO, not entirely, but in a France way. No more US nukes on British soil, though I doubt Britain would ever give up thier own. Anti-American hostility, obviously, but it would focus on the idea that the US is too irresponsible, not that they are evil or anything of the sort, so like OTL but moreso.
The real divergence comes if the plot is ever discovered. Would Britain be itching for a war? What if it was found out a decade later? The US would obviously want to avoid a conflict.