Very little info to go on only that it starts to blow cold air once the temp gauge is in the middle. It would help if you gave a model year and engine, but since you didn't the only thing that we can do is take a Wild Ass Guess (WAG).

I don't think it's a heater valve that is stuck open since you do get cold air eventually. I'm guessing you have a variable compressor and it takes a while for the rotary screw to decrease the displacement or is it increase the displacement for the compressor to pump enough freon R134 or refrigerant as the people in the trade would call it to make the magic of AC working. The only real way to tell is to hook up a set of gauges and tell us what the low side and high side readings are when the system is turned on.

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2006 Infiniti G35
2001 BMW 3.0I E53 X5 Build date 08/2000 SOLD
Lotus Europa 1970 Destroyed by fire
Lotus Europa 1970 S2 Renault Powered
Lotus Type 52 1970 Twincam Webers Powered
PORSCHE 911 Targa 1982 The Garage Queen
Audi Avant donated to Kars for Kids
BMW 525IT Sold
Audi 4000CS Quattro Sold
Jensen Healey Lotus Powered Sold
Opel 1900 Sold
Triumph Spitfire 1971 Sold
Triumph Spitfire 1968 Sold
Plymouth "Cuda" 340 Six pack SOLD