All you really need for your scenario is;
Suits and masks to get out of the contaminated area. It won't be a very large area.
Don't bunker down. It may be that nobody turns up and you will have to move eventually and by then your gear may be contaminated, you will be tired, hungry and thirsty.
Somewhere to go to that isn't a gov shelter.
I would arrange with my kids that if anything happens you go straight to Aunt Hildas or something. A rucksack with the gear in if possible. Some folding money for a taxi or payment. A mobile phone even if they are not allowed one normally. A list of instructions that they have already had explained and at a level they understand.
Any normal family is unlikely to be caught up.Oneor two people maybe.
Ignore the above for an all out nuclear war.