How to test a water pump?

There's not really any good way to test a pump other than to start the engine and watch the coolant flow in the radiator. Once the thermostat opens you should see coolant flow if the pump is working. What is the issue you are having? Perhaps that would give a better idea of the problem.

https://www.2carpros.com/articles/water-pump

Okay, for that get a cooling system pressure test and a combustion gas tester kit (both are loaner tools at a couple chain parts stores, so you don't need to buy them). Do the pressure test first on a cold engine. You can also pull the plugs and look at them while doing the test. You want them to all be the same color, so if you have say, 7 that are all tan with a bit of soot and 1 that is white and clean, you have a bad gasket in the clean bore. 2 very clean would show a gasket failure between them.
By doing the pressure test and plugs cold you have a head start. If everything there looks good and the plugs are all the same, start it up and do the combustion gas test. If it's a very slight leak or takes a long time to change the fluid color, you could try some Blue Devil in it.

The pump would leak outside the engine not into the oil.