The circuit operates by charging the capacitor C1 trough the base of the transistor. As long as the voltage over the capacitor is not too large, your transistor will conduct. When this voltage is larger than 24 minus 0.7 volts, the transistor will stop to conduct. Ideally it would stay this way. However, trough paracitic resistance in your capacitor, it will slowly discharge and finally (aparrently after 2 minuts) the transistor will turn on again.
You can however do the following:
You can make a circuit that turns on a transistor for some seconds when it is powered up. You can best connect the 24V supply to the same power source as the device you want to reset. When the circuit is turned on, your device will automatically reset.
For example, you could do it this way:
simulate this circuit – Schematic created using CircuitLab
Here R1 and C1 make the time delay until transistor 1 is turned on. When transistor 1 is turned on, transistor 2 will turn off and your relay stops actuating the reset button.
If you don't want to use the second transistor you could even place your relay at the location of R2, getting this circuit:
simulate this circuit
But in that case you would need a relay with NC contacts and the relay gets powered all the time (which isn't very power efficient of course).