When the tub is filling with water, it is very, very slow. I have checked the water flow from the supply lines, and it flows freely, hot and cold.

Undo the hoses at the machine and take a look into the unit's water valves.
There is a screen filter built into each valve. Check/clean it.
Do not remove it as it is there to keep crud out of the valve which could hold the valve open and cause a flood.

If you just removed the hoses at the taps and checked the flow, recheck it with the hoses connected. Some hoses also have a screen filter built into them. Also on rare occasions hoses can collapse intrnally restricting flow.

If it is a clogged hot side screen, it is a good idea to drain some water out of your hot water tank. Crud collects over time in the bottom of the tank. This will also improve the tanks efficiency.

If all OK then as applianceman said you will have to replace the valves.