My understanding is that the overwhelming majority of warheads belong(ed) to the Americans and Soviets, but what was the relationship to the smaller allied arsenals? Did Washington and Moscow have an itemized list of targets they expected/required their allies to hit or was the expectation that the two primary powers would do the heavy lifting and their allies were basically irrelevant?
The US only coordinated with the UK and France with regards to NATO nuclear weapons use, and that was conceived of primarily as a tactical use of nuclear weapons. So that would be about trying to have some kind of battlefield use of nuclear arms. How well that would have worked in practice, nobody (thankfully) knows, but questions about how to do this kind of "coordination" in the heat of battle took up a lot of anxiety regarding NATO and its nuclear weapons, and there were (and still are) annual exercises meant to help people understand how this would work.
I know of no coordination on strategic targets. It would not be very likely for two reasons. One is that the US regarded the specifics of its war plans to be very, very high levels of secrets, and would not have shared that information with a foreign nation. The other is that every nation involved here considered themselves to have an independent strategic deterrent. To coordinate a strategic deterrent would be to imply that ones forces were not independent (which would be a serious prestige issue, esp. for UK and France), but also would imply a reliance on other nations to "fill out" the full deterrent effect. There is just no situation in which, say, France would want to be dependent on the US to carry its "half of the bargain" on a deterrence threat with the USSR (and certainly not vice versa). Now, one could ask, doesn't the relative smallness of the French and UK arsenals imply that they would expect that a) most Soviet warheads would not be aimed at them, and b) thus that their real deterrent effect implies entangling the US into the conflict, and that might be true, but there is a big difference between knowing that is more or less how it shakes out and actually coordinating targets.
In terms of the Soviet Union and People's Republic of China, by the the time the PRC had nuclear weapons, much less a nuclear arsenal capable of deterring the United States (which is much later), the Soviets and Chinese were not allies (it is after the Sino-Soviet split). In fact, the PRC arsenal was in part meant to deter the Soviet Union. So they definitely did not coordinate targets, or anything else along these lines!!!