Here is a hack at what I think you asked for:
In the process I think I came up with a novel way of using a 555 as a positive-edge-triggered one-shot. Never built it, but am confident it will work as simulated.
In the simulation, the signal V(timer) [Green Trace] is a proxy for what the AC wall-clock timer does, except for convenience I made it be on for 120s and off for 120s instead of 12h and 12h, respectively. I'm assuming that the AC wall-clock timer controls a relay, which provides a contact closure as long as the AC wall-clock timer output is on.
That NO contact is shown as "S2" and "relay" in the schematic. It switches the left end of the timing capacitor C1 to ground while the AC wall-clock timer output is on, triggering the 555, initiating the timing cycle of 15sec as shown by V(rc) [Red Trace], and pulling-in the 555 output relay as shown by I(L1) [Blue Trace]. When the AC wall-clock timer relay turns off, diode D1 clamps the voltage at node "rc" one diode drop above Vcc.
Note that as long as the AC wall-clock timer duration is at least 15s, then it doesn't matter how long the duration is, even if it lasts 12h. The 555 output pulse duration is slightly longer than R1*C1.
C2, C3 and C4 are required bypass caps. D2 is a snubber for the output relay. As mentioned earlier, S2 is the NO contact controlled by the external 12h timer.