What is the simplest way to introduce a ~ 1 second delay in powering on a 230v AC device?
simulate this circuit &ndash; Schematic created using CircuitLab

In schematic above, you can choose to employ switch 2 (you can use a wall mains switch) or switch 1.
The values of R1 and C1 are indicative, C1 is charged to 3V in about 1 second with these values, which should be around the threshold voltage of NMOS M3. You should base the exact values on the threshold value of M3.
When M3 is triggered, it turns on M2, which turns on the relay. The relay turns on light controller 1 and the second 5V adapter.
The same circuit should be connected to this second adapter, so it turns on light controller 2 and adapter 3 after 1 second delay. And same circuit to adapater 3, which turns on light controller 3.

Most important of using a timer based on a R and C is that the capacitor is not being sourced or drained by something else rather than by the intended components (at the node indicated by the red arrow).
Also still in this schematic there is a leakage current, which draws away charging current: the gate source leakage current of M3. Although it is probably in the order of nA to uA, it may matter as the charging current is in the 0..10uA range. (With the edit, D2 also has leakage, but is the order of nA).

EDIT
If the discharging of C1 would be an issue, you could add a low leakage diode D2, which discharges C1 through R5 when the 5V adapter is turned off as shown in the circuit above.