Hmm.. I had new compressor and condensing radiator on my MK5 which was done under warrenty not long after I bought it.
The compressor had a distinct louder than normal whining sound that was linked to revs which i noticed from inside the car ( told its compressor internals disintegrating and metal shavings clogging the system so had that changed and AC system cleaned through).
Later my AC gas escaped and I only got the refrigerant low fault code. Compressor refuses to cut in also ( so it doesnt get damaged?) .. Dealers regassed and it escaped in 24hrs. They then regassed with a UV tracer which found a pinhole in the condensing radiator which was swapped out and a stone guard added for protection from the road.
So back to your story and enough of my past troubles.. You need gas for the compressor to kick in.
If you regas and the gas stays in and compressor kicks in and you get cold AC then jobs a good'n
IF you regas and gas stays in and compressor isnt working (faulty compressor or pressure switch/sensor) or sounding rough (failing compressor).
If you regas and lose gas quickly, but compressor works fine briefly with none of the above issues you have a leak in the system... Get a UV tracer pumped in with regas and use a uv lamp to examine all of system components and parts.
Its a long winded process but methodical and will help pinpoint the issue and not spend ££ unnecessarily!
Hope that lil essay helps you