It varies per brand, in addition to other factors, such as usage pattern, storage, and environment. For instance, the current formulation of Nancy Boy creams (without parabens) have been reputed to go bad in as little as six months, although I've had a tub for almost a year that is still good. Other brands seem to stay good almost forever. I've got some Trumper and Coate's tubs dating back to 2005 that are fine.
Another issue with older creams is dehydration. While some old creams of mine are practically as good as new, others have dehydrated to a point where I have to use them like shaving soaps - an approach that, while hardly ideal, actually works, and allows me to use up creams I might otherwise have thrown out.
Kept sealed and in a cool place, creams should be useable for many years. Some pots may need to be stirred before use, or loaded as a croap as noted above, but still produce lather. While creams will not have a shelf life of a soap I would think that you can count on a few years before things may go wonky. I just finished a pot of cream that was around 5 years old and it suffered no ill effects.
I couldn't tell you how creams smell when they go bad, as I've never actually encountered a bad cream. I've had a number of dehydrated creams; I have read reports of current formulation Nancy Boy creams growing mold; and I've come across a few balms and pre-shave oils that have gone rancid. But I guess I've been lucky with my creams.
I have tubs of TOBS creams that are upwards of 10 years old and are still in good shape. Some are a bit hard and have to be treated like soaps, but they still shave just fine.
I had a sample of Cella that had a smell like playdo. It lathered up well, but I just didn't like the smell. I have so many samples of ther creams that smell good that I didn't see the point of keeping it.