How to test hot tub grounding
I tried taking a multimeter to test for continuity between the house's
  ground and the hot tub water, but it didn't show anything. 


You should not see anything if the ground were properly connected. Most of the time, the hot tub is fiberglass and the piping is plastic, so there should be no connection to ground, EXCEPT via the pump housing, assuming a metal volute (the chamber part of the pump) and a proper grounding conductor on that pump housing. So showing no voltage between ground and the water means they are are the same potential, which is what you want.


  If I test for voltage between a phase and the hot tub water, I get
  240V.


As you should in your system, because the neutral of your 240V TT system is connected to ground somewhere and so is your water (again, through the pump volute and the pump body being properly grounded), so measuring between those two points should show as the full 240V.