I'm by no means an air conditioning expert, but it shouldn't take very long for the expansion valve to initially let the gas through, until the compresses it and the condenser turns it into a liquid. Since car AC systems are small, this should happen pretty fast.
Now the AC may be blowing cold air, but until the hot air in the car is replaced, the car will still be hot, and that can take a few minutes.
- If you're recircing air, the AC is trying to cool the very hot interior air, try turning recirc off to cool down the cooler outside air first for the first few minutes, then turn recirc back on.
- Try cracking your windows so as cooler air enters the car, it settles down lower and pushes the lighter, hotter air out the windows.
As someone else wrote: "How long before a standard car's typical A/C will start to cool your car? The discharge air temperature should start dropping immediately, and after about a minute it should be cool (the A/C has cooled off itself and is now starting on the cabin air). After two minutes the discharge air should be very cold, (car A/Cs cool the discharge air to a MUCH colder temperature than systems for your home or office)."
The first 4 results should help:
https://www.google.com/search?q=how+fast+does+a+car+ac+cool+down One of those results even talks about related mechanical problems which could be affecting the AC.