Shower pump sometimes not kicking in.

There shouldn't be any issue linked between the Boiler/heating and pump. If you turn off the boiler and your heating I'm sure the issue would persist. The only thing short of the pump failing is air in the pump. In which case switch off the pumps electrics at the wall and run the shower for 5 mins to remove any airlocks. Or If you know how to isolate the pump using the isolation valves on the hoses going to the shower pump. Alternatively if your not very confident switch off the main stopcock and test the shower to ensure the water is actually off. Then unscrew the hoses at the pump, check the mesh filters, they maybe blocked. remove clean them of debris and put hoses back on the same way with all the rubber washers

The problem is air getting drawn into your pump when the heating is on, an Essex flange fitted on your cylinder will eliminate the air and solve your problem.

The domestic hot and cold water should not be effected by the heating. You are best to ensure that the supply to the pump meets manufacturer specifications, ie there is a flange at the draw off point at the outlet on the cylinder. We find these are rarely installed and they should be to avoid the drawing in of air from the vent on the hot water cylinder.

Your hot water heating coil inside your cylinder may be getting starved of heat when your radiators are on as there is more load going to all the rads instead of the hot water. This would mean hot water would take longer to heat. This is easily fixed with the system being correctly balanced by a heating engineer.