Mineral oil is more viscous than water. It also has a considerably lower bulk modulus. If you apply a stress on the fluid, in this case pressure, you can expect the volume to change as a result of the bulk modulus.
You haven't told us the volume of the fluid under pressure, or the time taken for each fluid to stabilise. We only know that oil takes longer to stabilise and that is expected.
When you apply a pressure to a fluid, its volume must not change if the pressure is going to rise. Air has a much lower bulk modulus and its volume reduces to maintain a constant pressure. In liquids, the bulk modulus is much higher and so, when the force is applied to increase the pressure the volume does not change by as much, hardly anything.
In a hydrostatic tests, the internal galleries, areas and zones are filled with fluid and that fluid moves from high pressure areas to low pressure areas. All of the threads on the fittings, all of the seal housings, all of the seals, everything has to undergo strain to some extent to see the pressure rise on the system pressure gauge.
With water having a higher bulk modulus and lower viscosity, it will be more resistant to pressure and it will flow into the gaps and galleries more easily as the system is put under stress or pressure. If you shut of the system when the required pressure is reached, with water, it can be assumed that nearly all of the fluid has reached all of the threads and empty galleries.
With mineral having a higher viscosity and lower bulk modulus, it will absorb some pressure and will resistant to shear and will not flow into the gaps and galleries as quickly as water does. With rising pressure that is halted when a pressure switch is triggered or stopped manually, the fluid is still expanding and if the stress input has stopped, it can be expected that the pressure will decay as the fluid in the system is still moving from high pressure to low pressure and filling the all of the gaps.
If you want to improve the stabilisation of the pressure, perhaps use a fluid like paraffin or kerosene. It will be more like water in terms of the pressure rise and pressure stability. If you really only want to avoid the oxidisation with using ferrous materials with water, then perhaps use a water glycol mix? Heating the system up won't make much difference as the viscosity index of the mineral oil will compensate for the heat increase. You could also leave the pump on for longer to help to bring the entire system up to pressure.
Some details of the pressure rise rate, pressure decay rate, flow rate, pressurised volume details etc will help to give more accurate help and advice.
