Sorry, been having computer trouble.
I have reworked the entire input section of the timer. Now it should not trigger when power is applied to the alarm circuit, only when the alarm is supposed to trigger it. Alarm_Vcc should be connected to the power supply of the alarm circuit, while Alarm_input is the actual signal from the alarm.
The circuit now has two start up timers, the original for the timer, and the one connected to the 555 which only starts when the alarm is first turned on. It also has the peak detector that is built into the 555 itself. This should prevent multiple triggers while the RC timer is still low.
If anything seems like it is taking too long to activate, then just make the capacitors smaller. If circuit is still giving issues then make the capacitors larger.