It would work, but it's unnecessary
Deep layers in glaciers and ice caps can quite commonly be uncontaminated by surface activity for hundreds of thousands of years, and in a few rare cases they are over a million years old.
Mining into ice is quite easy; keeping the tunnels open is much harder, but not impossible. Because of regelation and the general plasticity of ice close to its freezing point, the ice will tend to gradually flow into the tunnels and close them. This could be prevented with a tunnelling shield. In practice for a material like ice, recovery is likely to be much more efficient with something like solution mining using superheated steam, a technique that completely eliminates the complications and dangers of tunnelling.
However, none of this is necessary unless the nuclear winter has turned the entire world into a (cold) desert. Potable water is not a finite resource; it is the ultimate renewable resource. It is constantly renewed by the water cycle, which purifies water by distillation. After the few days or possibly weeks required to wash fine radioactive dusts out of the upper atmosphere, any rain or snow falling will be potable. All that is necessary is a way to collect it without it getting recontaminated by contact with soil.
On a small scale this can be done with roof collection. Heavy snow makes it easier: the few inches in contact with the ground will be suspect, but the rest is fine.
It will be much more difficult to collect on a large scale, but not impossible. In high run-off impermeable terrain such as granitic mountains, much of the hazardous material will be washed away in the first few heavy melts. Each subsequent melt will still have some, but the level will drop rapidly with every melt and soon you will be able to collect potable water from gushing mountain torrents on a light industrial scale.
A particular advantage here is that, more than almost any other type of contamination, radioisotopes in water are especially easy to detect and measure, so any source that may possibly be clean can be quickly and reliably assessed.
What if there's no precipitation?
Maybe you have the type of nuclear winter where the entire world is ice-covered. It is too cold for any surface moisture to evaporate, and so there will be no precipitation -- no rain, and no snow either. A global ice desert. This is essentially what Sagan et al. proposed with the original nuclear winter hypothesis: that the smoke of ten thousand burning cities and forests would blot out the sun long enough that the surface would cool so far, that runaway albedo causes glaciation.
This theory is not now regarded as at all likely to be true -- a nuclear war will not trigger a "snowball earth." However if it did, finding potable water is the least of your troubles. No plant life will exist anywhere on earth, and when your tinned food runs out, everyone will die. Actually, you will probably freeze to death long before then, because it's going to get really cold: somewhere in the ballpark of -150°F ... at noon at the equator.