Here in the UK, a network of bunkers were built and maintained into the 1990's - mostly by the MOD.
Unfortunately for us plebs, the vast majority of these bunkers were not intended for use by the general public. Instead the bunkers were intended to be used by local council members, police chiefs, government ministers, military personnel and, of course, the Royal Family.
More impressive were about 30 huge regional bunkers, each capable of sheltering hundreds of people. Here it was hoped that key MPs, police chiefs and other VIPs would be able to retreat to begin the task of fighting back and rebuilding the country. - Express.co.uk
That isn't to say that a number of ordinary people wouldn't have been able to gain access to these regional bunkers in case of attack, the largest had room for 6,000 people, but the vast majority of us would have been left to fend for ourselves.
A series of leaflets, radio broadcasts and public information films were circulated throughout the 1970's and 1980's called "Protect and Survive". These instructed the general public on what steps to take should the UK have been attacked.
Examples can be seen all over the web with a simple google search, but the advice included building shelters using cushions, and closing windows. Steps that would have given you something to do, but not have provided any long, or most likely short, term protection.
Seeing as, for the most part, the general population was expected to basically fend for itself until the regional control centres could begin to enact the various plans (that are likely still classified) it seems unlikely that pets were accounted for at all.