I grew up in the 60s and 70s. I remember some of the cold war films they showed us in school. According to them:
Stay indoors. If you’re close enough to the blast, you’re probably dead anyway (that was mostly just implied). For many more people, though, fallout is going to be the main worry - so stay inside. And even if you still have running water... you probably shouldn’t drink it. Use what’s already in the back of the toilet and in your hot water heater.
Of course nowadays, post 9/11, most reservoirs are supposedly covered - so I have no idea if that’s as important.
In any case, water is probably going to be the main short term issue. If you have some pre-blast warning, filling up as many containers as possible with water is a good idea.