My dishwasher begins its cycle with a whirring noise, what sounds like the powder drawer then clicks open, but after that nothing happens. The water does not come in. The dial clicks round a couple more points, the machine 'whirrs' again, but still no water/washing action. The machine then just stops/cuts out. Is this a plumbing problem or an electrical problem?

The whiring noise is the machine giving itself time to fill with water before it kicks into the next cycle stage, if the machine sensor doesnt detect water within the time set the machine will not go to the next stage and will detect/display a machine fault/error. Can you hear water flow into machine while hearing this noise also? If not This would suggest water intake to machine has been interrupted. if the water flow is not interrupted then with the soap not dissolving it could be either temp thermostate or pump blokage. You'll need your model number to search for the machine Service manual online if you don't still have it. *Do not attempt to clear pump/ diagnose or open your machine without turning machine off and disconnecting plug from the wall*

It sounds like the water pump has seized judging from the noise that you were hearing.
When the water seals become old and worn, water seeps through and ends up in the pump motor bearings, which corrode and eventually fail.
Open the machine up and check the pump, they are not repairable, will have to be replaced should this be the case.

All power and sounds should stop when any dishwasher door is opened.
Your door switch may be stuck causing it to continue.
The 90 second sound at the beginning of the cycle is the water pump extracting any residual water before taking on clean water for the wash cycle.
Any excessive noise from the drain pump could be debris in the pump. e.g. broken glass.
You can remove the pump and check it for blockage or any foreign object.

The Problem is Related to the Dishdrawers Motherboard Control Center This usually happens when water condensation sits on the board to long and Short Circuits controls.

Both problems sound like motor has failed or is failing.

I doubt that the user manual covers de-liming the Hobart Dishwasher.   If I owned this (fine, expensive) dishwasher, this is what I would do.
1.) Discard any generic, ultra-cheap detergent.  Stop using gels or tablets and only use powders.  Buy a box of brand name powder (like Electrosol or Cascade)  You will be amazed how much difference that will make because the newest generics have mineral bulk added that build up faster then hard water does on both the stainless steel enclosure plus the silverware and even plates.  White patches.  I LOVE generic products and saving money, but low end dishwasher powder is one place I no longer choose a generic product.
2.) Add two cups of white vinegar to the empty dishwasher and leave it there until the morning.  Run a quick cycle with no detergent added.  The vinegar will be pumped out immediately after you begin this quick cycle.
3.) As soon as the dishwasher fills with water during that quick cycle and begins a wash cycle, add two cups of white vinegar and close the door and allow to run the full wash cycle.
4.) When the dishwasher finishes the drain cycle and begins to fill again, open the dishwasher and check the enclosure for remaining "lime" deposits.  If they remain, keep adding 2 cups white vinegar each time the dishwasher refills and begins to wash/rinse again.
You should be delighted with the results. Long term, you should consider adding a whole house water softener and then reduce the amount of dishwasher (and laundry) detergent by one half.  Amazing increase in quality of life by adding a water softener.  You will be delighted.
Please post back with you results, OK?  Good Luck!

The soap dispenser drawer is supposed to open during the cycle. Try first to understand if the drawer is opening or not during cycle.

If the drawer is not opening, then check if the opening mechanism is stuck, if it is blocked by dirt and soap, if it is broken, or if something, for examble a basket too close to dispenser drawer is preventing the drawer from opening.

Some dishwasher works only with powder, in that case indications are given on the owners manual.

If the drawer opens and the table, or powder is still staying stuck in the drawer, then that means that the spray does not have enough power to solve the tablet.
In that case check spray arms and impeller pump.Also check water temperature.

If the problem is also happening with powder, then it is probably a drawer not opening on cycle.

If it is not overfilling, and then leaking, the whirring noise can be a defective water inlet valve, that stays stuck on open, see here : It's overfilling

If it is not overfilling, the two problems may be unrelated, and the noise can be froma foreign object inside the object trap.

See here about leaks and noise

It's noisy

It leaks

The pump motor is beginning to seize up. You may be able to take the dishwasher apart and lubricate the bearings and keep it going for a while, but at some point you will likely have to replace the pump motor assembly.