Well - i would concur that the air conditioning is not working. Was the clutch pulling in on the aircon pump ... how much R134a was in your system before they 'refilled it' - they will know this because the machine will weigh it on the 'evacuation process'
Yes a dye is used - generally it comes in with the oil ( PAG or equiv ) and this will show up under the UV light to show any leaks.
You rellay neeto start at the compressor pump in your case and work backwards. Select aircon to on and advise wether the compressor pump is running - visual check LH side of engine driven by belt. You will see that the end of the pump is turning with the belt. If it isn't then the system has detected a problem and shut off - or there iss not enough or too much R134 in the system, ot the Clutch relay is defunct, or the Fan relay is burned etc etc bearing in mind that the aircon panel is also integral to the system.. Report back please.
Eddie