Water is Always the Big Post Apocalyptic Problem

 Storing or transporting water enough for a couple weeks is always a problem.  Covered wells are fine, but do you have a pump to pull it up.  My old house had a 1000 gallon gravity fed water tank.  Any moving water is likely to be contaminated by fallout for weeks or even months (for the low grade high altitude fallout).  

The water itself is not a problem.  Neutrons from the fallout will produce a small amount of deuterium (stable and safe) and O17 or O18 (both stable).  Precious little of both actually; the nuclear cross-sections of hydrogen and oxygen are tiny.  The risk is the fallout in the water some of which is water soluble; filtering is not enough.  However, a solar still is a solution.  You will get pure and somewhat flat tasting water with no contaminants.  I see a variety of distilling rigs on Amazon, generally a bit pricey.

I think you could make your own a bit cheaper.  When I was young, I remember reading instructions for using that awful plastic car upholstery for distilling water out of the air if stranded in the desert.

Both of these assume a limited supply of either ground water or sea water.  I may try this in the next day or two.  Cold weather is not a problem; clouds are.  The sunlight warms up whatever water you have even if it is glowing.  Cold air above the plastic speeds up liquifying the water vapor.