In my intuition one would want the "ordering" of the voltages to be correct during the complete time of the startup transition.
So when switching, I would recommend to first power the 15.7 V then 5.2 V then 3.3 V so that there is never a reverse voltage between the different supplies.
If there are diodes from each lower to the next higher voltage, then it makes sense.
But it totally depends on the circuit. If the analogue section (+/-15V) is driving ADCs in the digital section (probably +5V), then it may not like it if the input signal is present before its power supply voltage.
I also remember some Epson display controller ics, that need the lower core voltage first before the IO voltage, otherwise they will get quite hot:
"When applying supply voltages to the S1D13513, Core VDD must be applied to the chip before, or simultaneously with H VDD, or damage to the chip may result."
If you don't know any specific requirements, switching everything on at the same time is probably the best solution.