The reason I asked about the flow of water and air, if there was a lot of air intrained in the flow, it could indicate an air lock which might have affected the pressure switch, or a low flow would have indicated a restriction, but you have told us the flow is normal
For some reason the water pressure is insufficient to trip the pressure switch. The question is why?
Because the flow jet pumps use a diaphragm construction, they are positive displacement pumps, and they can produce some very high pressures. This is why they have a pressure switch to control them. But being positive displacement pumps it means if it's running, it must be delivering water, it's got to be going somewhere. Have you checked for any leaks?
In particular caravan water heaters have pressure relief valve (PRV) which normally dump excess pressure water under the caravan. Depending on the model of water heater, the PRV may also be drain valve. Have you forgotten to close the drain valve after draining for storage?
Or is it possible the system wasn't fully drained before winter storage, in which case if there was any water in the valves, and it froze, the forming ice can damage taps, valves, pipe joints and pump.
If you have no leaks, then it points to the pump's internal valves not closing properly. This could allow the water to flow back through the inlet pipe for the pump, preventing pressure build up to trigger the pressure switch.
Do you have a water filter? These are usually made with activated carbon granules contained by a filter paper or pouch. If the filter is to old or has been damaged, they can often release the granules into the water flow. The granules can jam in valves causing them to leak, but you would normally have seen black granules in the water from the taps at some time.