If I have a 15cm diameter by 3 metre high water tank with a non-return one way valve at the bottom attached to a water pump, what size and power of electric pump would I need to pump water to fill the tank?

Thanks

You did not specify a flow rate so that doesn't come into the analysis.  I think since the flow rate can be low it is ok to ignore head losses.  So all you need is a pump that can develop a delta P of about 4.3 psi.  That is a tiny pump.  I looked at Amazon and you can get a 12v DC pump for about $19 that would do the job, it is actually over powered.

Thank you. It is a bit of chicken and egg situation regarding the flowrate and I will only know it once I have carried out my experiments.