Water Flow

Is there a simple way to measure the flow of water inside heating pipes?

Yes. In a closed loop hydronic heating system the circulator pump characteristic flow curve can be used to compute flow. You measure the pressure differential across the pump with a pressure gauge at the inlet and outlet of the pump. The pressure difference is a measure of the piping resistance (pipe head) and gives you the operating point on the manufactures pump curve. You then read the flow (GPM) from the curve. What is your circulator pump make and model? I'll post the curve and the measurement procedure.

The direct method to make a flow measurement requires a flowmeter. It's usually a turbine like wheel in the flow path with electronics to measure its rotation. It's way to expensive for residential market use.