Why is it taking so long for my A/C to start?
This first thing to try is removing the thermostat and shorting the R and G wires. Usually, these are the red and green wires. This should start up the fan in the blower. If that works, try shorting R and Y (usually yellow). This should start up the fan in the blower, and the AC compressor. If all of this happens immediately upon shorting the wires together, then the thermostat is likely failed.

If shorting the wires as described does not immediately bring on the blower or compressor, the problem is probably either bad wiring or a bad relay or control board in your air handler.