Is "mS" micro sec or milli sec?
mS is the standard abbreviation for millisecond. µS (or more often uS on this forum) denotes microsecond.
The time it takes a micro to do a cold start depends on many things: the micro type/complexity, the clock rate, the peripherals which need to be initialised, the initialisation code efficiency....
I've just measured the time one of my micro projects (MSP430-based, 1MHz clock) takes to initialise as 287uS.
No appreciable delay needed
To confirm this, I simulated the power supply arrangement which might be used in my suggested circuit. The sim showed that the 9V regulated output is stable after about
5mS. Add to that <1mS for the micro to initialise.
In the sim the yellow curve is the 9V output, the blue curve is the nominal 24V AC from the transformer, the red curve is the voltage on the capacitor. The LT1086 is a variable voltage regulator with its output set to 9V by resistors R2, R3 (I didn't have a model of the LM7809).