Well, if you have had water leaking then sounds like time to solve that first. If you can't solve the water problem economically and it is a rare occurrence, then your solution is (when the concrete is dry) is to go with polished concrete or permanent finished concrete (epoxy or polyurea or stained) which will still be cold, or perhaps a totally synthetic fiber (including backing) area or throw carpets or runners which can be picked up and dried out when they do get wet. Basically speaking, sealants and vapor barriers can take care of minor vapor transmission (flooring which does not show dampness basically) which can damage full-coverage non-breathable flooring, but not "water" or "wet" flooring - cover it and the flooring or framing WILL eventually get wet and moldy or fungusy (imagine mushrooms fgrowing along the edges of your carpet, for instance - I have seen it several times).
Any sort of laminate or plank or sheet flooring, unless the water problem is permanently solved, will result in mold issues and possible (due to spread of water under the flooring rather than largely evaporating as it likely does right now) may also start causing deterioration of framing wood and corrosion of utility runs (wiring or metal piping) as well as possible rot/mold in the walls.
Tile works because first the grout and thinset (if right type is used) is portland cement based so actuyally gets stronger with wetting, and the tile is (excluding "clay" tiles like quarry or many "Mexican" tiles) waterproof, though as you say it is cold.
You can find a fair number of previous questions with answers about wet/damp concrete flooring issues in the Home > Flooring and Home > Basement Waterproofing links under Browse Projects, at lower left - scan through the question topics for ones related to types of flooring to use on concrete floors.
My recommendation - if dampness can't be permanently solved go with throw or area rugs small enough to be easily pulled up and dried out, with all synthetic fibers to reduce the mildew/mold risk - or buy yourself some fuzzy slippers or mocassins to keep the feet warm.