Filtering Radiation Contaminated Water

First of all, all bottled or sealed water will be safe from contamination even if they receive radiation. Water itself (H2O) doesn't go radioactive, but the stuff that are dilluted in it can be radioactive. If the water was bottled before fallout reached the springs, then it's going to be clean.

What we're worried about here, is contamination with radioactive matterials (fissile matterials), like Caesium, Strodium, Iodine isotopes, etc.
We've two ways to go about filtering these contaminants out:

Good old fashioned distillation. This is applicable if you can construct a distillery like the ones used for moonshine. It will work best with filtering out heavy isotope conatminants like Caesium and Strodium, but not for hydrogen isotopes. However hydrogen isotopes usually come from leaked nuclear plant cooling water, too close to ground zero, so you won't likely pick up water in those areas anyway. Make sure you keep the distillery clean, don't let the sediments build up because radiation will corrode the metals and in the case of an accident with the distillation, released steam can upturn the sediment and you'll breathe it in.

Reverse osmosis and activated charcoal filters. According to the EPA

However it can't remove gaseous contaminants like radon.
Activated charcoal works by absorbing various soluble matterials in the water and air. It can absorb not only heavy metals but also inorganic and organic (biological) chemical contaminants. They are primarily used to filter ventilation in medically irradiated chambers.