I'm no plumber, but I would imagine that (if the hot faucet is giving you cold water) that the hot and cold pipes leading into the bathroom have been somehow switched.
Odds are that the cold water lines to the faucet you are trying to use are shut off but the hot water lines coming from the water heater are not. That is why you have hot water but no cold water.
Because hot water is still in the pipes from when you turned on the hot water, and when you turned on the cold faucet it had to let out the hot water first. This means you have a single-pipe sink. Double-pipe sinks have a pipe for hot and a pipe for cold and it blends them as it turns on.
If you are not living in the desert where the cold water is hot during the summer, check to see if perhaps the plumbing is backwards: is the hot water faucet giving cold water? I once had hot water in my toilets.
Assuming that you tuned the faucet to hot and let it run until the hot water should have reached it, you may want to check your water heater and make sure that it is working.
The faucet opening screw thread is stripped, the faucet washer has collapsed, the pipe is furred up, the boiler has a problem
If water is only drawn from the cold faucet, then no, the water heater is not involved. However, if water is drawn from the hot faucet- even for a few seconds- hot water IS withdrawn from the heater, and fresh cold water drawn in to be heated- even if the hot water did not make it all the way through the pipe to the faucet- and the water heater will work to heat that fresh cold water.
This occurs when your hot water heater is either broken, disconnected, or has exhausted its supply of heated water. In cold weather, heated water may be cooling in the pipes on the way to the faucet, especially when it has just been turned on.
Do you have hot water at other fixtures in the house? If not you have a water heater problem. If yes the Shower valve needs to be rebuilt if it is a bressure balanced faucet the concentration should be on the pressure balancing spool area of the faucet.
Kitchen faucet? If so, the sprayer attaches to the underside of the faucet in the middle.