This is definitely puzzling. The water system in our trailers is actually very simple in how it works.
When connected to city water the cold should work fine as there are no valves on the cold water main line unless one was installed by an owner for some reason. I have nothing at all on my cold main line.
When using city water, Donna is correct that the pump should, nor need not, be used. If needed, disconnect city water.
Even if there was a wrong flow setting on the water heater valves it would not affect the cold as mentioned above. If water is flowing through the water heater and hot water coming out of the taps the valves are set correctly.
If any water is working when connected to city water and the pump turned off then there is no issue with the check valve at the external connection.
If all is working as written then the issue has to be with the cold water line. I am at a loss to what it could be as even a wee bit of debris will not totally block a line, and I really doubt it is that if it worked before. If there is a kink in the main line it is almost impossible that it would stop all flow as not only would that be hard to achieve.
Once you have proven hot water to all taps, have you checked all the taps for cold supply, the kitchen and bath sinks, toilet and outside taps? Leave them open for a short while and if there is air it should clear quickly and you will get water. Where I am going with this is thinking that maybe one tap/valve on the cold water circuit has failed.
I really wish I was there to evaluate, that would make things much simpler in my wee mind.