I was thinking of the Thirteen Days movie with Kevin Costner about the Cuban Missile Crisis and there is a scene there if i recall correctly when Costner (president's chied of staff) visits the USSR embassy at the height of the crisis and the secretary is terrified, basically trembling because of the prospect of nuclear war. Would a country notify its own embassy staff before launching a nuclear strike again against a city that hosts the attacker's embassy? Or would the staff be considered expendable in order for the suprise to be total, maybe prevent leaks, whatever?
This isn't a direct answer to your question (which, for the reasons incendi pointed out, is likely impossible), but during the Cuban Missile Crisis the Soviet Embassy in DC was kept very closely abreast of the developments in Cuba and the posture of Russian nuclear forces. Obviously, this is because a complete picture of the situation was necessary for them to be useful in negotiating with the Americans. While it's not necessarily true for every situation in which the two countries might have bombed each other, I think it's reasonable to assume that the US embassy in Moscow would have been similarly closely appraised of any military/diplomatic situations which had the potential to escalate to nuclear warfare.
Of course, this doesn't necessarily speak to a spontaneous surprise attack, and it's mostly guesswork and analogy anyways. Without good sources- like actual policy documents from that era, which are mostly still classified- we probably can't know.
I don't know of official policy on this. Obviously they would want their embassies somewhat in the loop on these kinds of things, because the embassies are presumably trying to play a diplomatic role. But you wouldn't want to warn your embassy about your own nuclear attack. Why not? Because if the entire embassy staff starts fleeing for the hills it will tip off the country in question that you are about to attack them. Embassies were closely watched by intelligence agencies, and in tense periods, all countries are on the lookout for evidence of an imminent nuclear attack. The Soviets, for example, parsed all sorts of data as part of Project RYAN in the 1980s, looking for any sign that the US thought war was about to happen (e.g. if certain types of fuel supplies or agricultural supplies were being hoarded). It would be an extremely alarming thing to intelligence agencies if embassy staff suddenly went into a bomb shelter or fled the country.