Hello Annie and Ted,
Let me try to clarify the symptoms you have, From your posting it seems you get adequate flow of both hot and cold water at the bathroom, but both hot and cold are absent at the kitchen.
based on my assumption, then the pump and the pipework is ok at least to the bathroom. The question is why is there no water at the kitchen.
My first thought is that you have microswitches in each of tap fittings that turn the pump on when ever the tap is opened. You can usually confirm this arrangement by checking each tap and listening for feint click as the micro switch responds to turning the fawcett on and again when you turn it off.
If that is correct, then it seems likely that either the microswitches in the taps have gone faulty (which does happen from time to time) BUT it is unlikely that both the hot and cold would fail at the same time, so my next thought would be to check the wiring from the taps to make sure a connection hasn't come loose or the cable has been damaged - That would certainly account for both taps failing together.
Now another tell tale symptom you describe is the fact you get some water when you turn on the taps but rapidly reduces to a dribble. This suggests to me the pipes are actually ok, and teh water you get is from the hot tank where a small pressure build up does occur and is stored by the air cap in the tank. The air cap is meant to be there to allow the cold water to expand as it is heated. However it does mean that when the hot tap is opened the air cap can push some water out which may well be the flow you describe.
There are two checks you can easily do:
You can usually just hear the pump running inside the caravan, but if not get some one to help by listening outside, then go round the caravan and turn each tap on and off and listen for the pump. Keep note of ant taps when the pump does not run. Those are the taps that need attention.
The second test is to fully open the kitchen cold tap, then go to the bathroom and open any other tap that you know works just a little. If you get water flow at the kitchen then you know the cold pipe work is ok, Do the same for the hot. This then proves its something to do with the kitchen taps switching mechanism.
If you still get no flow then the problem is more deep seated and you need to check the pipe runs for kinks or other things that might restrict flow to the affected taps.