How can I determine why all the faucets are giving only warm water when turned to cold?
Check the temperature of the main home water supply pipe, and compare it to other pipes in the house. Nothing else will be colder, but the other cold water pipes shouldn't be much warmer.

If you find a presumed cold water pipe that's significantly warmer than the main supply, then you can search for the source of the heat. If, for instance, the washing machine is bridging its cold and hot source, then perhaps your cold water faucets are pulling some hot water through the washing machine.