I copied the albeit cryptically translated instructions from what I think is the module you have:
P0--0: T1 Timer time mode is seconds
P0-1: T1 timer time mode is minutes
P0--2: T1 timing time mode is hour
P1--0: T1 time delay relay (T1 timer)
P1--1: T1 delay time relay releases (T1 timer)
P1--2: time delay relay after T1 (T1 timer), then the delay time T2 successor electrical release (T2 timing) ends.
P1--3: T1 delay time relay releases (T1 timer), then the delay time T2 successor Electric energized (T2 timing) ends.
P1--4: time delay relay after T1 (T1 timer), then the delay time T2 successor electrical release (T2 timer), repeat the cycle.
P1--5: T1 delay time relay releases (T1 timer), then the delay time T2 successor Electric energized (T2 timer), repeat the cycle.
As I understand it, your timers are set to one of the cycling modes (P1-4 or P1-5) turning on and off repeatedly and forever (as long as 12V DC is applied to them).
So, to your quest for a "... a delayed start of providing 12VDC power to one of these timers when I hit the on button…"
If you can spare one of these modules for the sole purpose of delaying 12V, then I'd think setting it to mode P1-0 with the T1 parameter settable in 0-999 seconds (P0-0) would give you the 0-60 sec delay set-ability. That is, in the classic 12V DC (e.g., automotive) delay timer application, you apply 12V DC when starting the engine and some time delay circuit waits seconds before applying 12V to accessories (lights, a/c, heater, whatever) so as not to steal precious battery "Cold-Cranking-Amps."
Another option if you don't have a spare timer module for your multiple-accessory animation would be to have the first module to come on power additional modules. So in your 3 circuit example, say the rotor comes on first, lights come on 10 sec after rotor, and landing lights 20 sec after rotor.
The rotor timer would be set up in cycling mode for, say, 5 minutes ON and 10 minutes OFF, turning on and off every 15 minutes forever. Then the lights module would be set for a Delay-on of 10 sec using mode P1-0. And the landing lights module would be set for a Delay-on of 20 sec using mode P1-0. The rub is that when the rotor turns OFF, the lights and landing lights turn off at the same time.
