The NASA doc above is the definitive source, as it is based on original research.
The important things to take from it:
- The formula only applies if the tyre can be supported by the fluid - that is that the depth of the water above the surface of the runway is greater than the tyre tread depth.
- The speed required to support the weight of the aircraft is (in almost every normal case) greater than the touch down speed of that aircraft.
... so the pilot "clued up" on aquaplaning understands that:
- it is worth noting the tread depth on the walk-around.
- That you don't need to worry about "aquaplaning" if the runway isn't flooded
- that regardless of the calculated speed, you always need to worry about "slippery" conditions if it is wet (or contaminated with rubber, or oil or slush etc.) and,
- that, if asked by a pedant, it is only the main wheel pressure you need to calculate - as the nose wheel doesn't (in almost every case) have any braking, and at the speeds in question, steering authority is still with the rudder.