what size is the larger plastic loft tank? (the round thing in the airing cupboard is not a tank, it is a cylinder)
It may have a capacity embossed on the side, in litres or gallons.
Lift the lid (it should have one). Is there dirt, drowned wildlife, bits of insulation wool at the bottom or in the outlet pipe? If there is no lid, measure the top of the tank with a tape measure, and buy a lid from a DIY shed or plumbing place. Note the brand of the tank if visible. Observe if the limescale tidemark is a couple of inches below the overflow pipe hole neat the top of the tank.
Observe the float. Look for a way to tie it up (often a nail or hook will be in the roof timber above, from last time) with a piece of string to prevent it filling the tank. Tie it up and run the cold tap to half-fill the bath. Then take a bucket into the loft, hold it under the ballcock, undo the string, time how long it takes to fill the bucket, calculate how many litres per minute it delivers. If there is a valve on the pipe to the ballcock, first turn it fully open then half a turn back. It does not matter if that makes the ballcock noisy, that's easily fixed.
Getting air in the pump will wear it out faster.