For what it's worth... my experience with car air-con where it blows cold for quite a while (say 20 or 30 minutes) then seemingly stops working, is that the air-con compressor clutch gap is worn beyond tolerances. And once the underbonnet temperatures rise about a certain point, thermal effects on the electro-magnetic actuation of the air-con clutch means it's no longer got sufficient umph to close the widened gap.
As a secondary note to that, low / lower gas pressures, previously, can cause the compressor to cycle quite frequently and cause more wear to the compressor clutch.
In most instances, you can check - if you've got reasonable access to the compressor clutch plate / gap and some feeler guages, but you'd need to know the gap tolerances.
If it is this, some people manage to do true repairs with it still on the car (usually involving removing the outer clutch plate, and removing a shim or washer - or grinding one down and putting it back), sometimes people manage to bodge solutions by being able to wedge suitable shim material on the outside of the clutch plate - forcing it in a bit. It will depend on implementation, though, because in some cases, the only way may be to remove the compressor (de-gas first) work on it, and put it back.
I've done this before, on other cars - well actually, I bought a used compresser from ebay, removed the clutch plate, ground down one of the shim / washers a little, replaced it all so the clutch gap was at the unworn end of the tolerances, then had the compressor replaced on the car (I then sorted out the shim(s) on the removed compressor, so I then had a good spare).
If you were having these symptoms, but now the air-con is not working properly at all, it could be that there was a slight leak, causing low-ish gas pressures, causing the compressor to cycle more frequently, causing additional wear to the compressor clutch, causing it to stop blowing cold after a period of running the car, to then the gas pressure falling below threshold to largely stop the compressor engaging at all.