Hello Intransient,
Let me try to describe what I think you may be experiencing. Initially you start to use the shower and at first you get some water for a second or two but the pump has not yet turned on. Then the pump turns on, and you start to get WARM water through. Then after a few seconds the pump turns off and you get a surge of HOT water, then after a few seconds the pump turns on again and the water returns to WARM. This oscillation between only HOT and mixed WARM water flows recurs until either you stop showering.
This effect called surging, and it happens because the pump is controlled by a pressure switch, and the pump is capable of delivering more water than is being used by the shower, and most important of all, the cold water system does not have a surge damper fitted.
The reason this happens is the hot water storage tank needs a small air cap in the tank to allow for the expansion of the water as it is heated. The hot tank also has a non return valve fitted at its inlet, so cold water can be pushed in but wont run back out. Air in the tank not only allows the heated water to expand, but it is also partially compressed when the pump runs. (This is why when you turn a hot tap off the pump usually runs for a few seconds as it recompresses the hot tank)
By comparison the cold system does not have a similar air cap so the cold water pressure response is much quicker to drop when the pump turns off. This means that during a shower if the cold system pressurises and turns the pump off, the cold water flow stops immediately, but the hot water continues to flow because the compressed air cap pushing the hot water out It will continue to push water out until its pressure drops enough to allow some cold water to enter the tank which in turn drops the pressure in cold line and turns the pump on again. And so the cycle continues.
Some more recent water pump systems incorporate some clever circuitry which is designed to modulate the pumps delivery to match the water being used. These should eliminate the surge phenomena provided it is set up correctly. The older pumps need a surge damper to be fitted to the cold pipework. This essentially adds an air cap to the cold supply to match the hot tanks air cap, and thus reduce any rapid pressure differentials between hot and cold supplies.