Pin 7 of the 1237 determines the turn on delay.
Default value per data sheet is 33uF and 56K ohms.
C2 on your board = 47uF, Resistor (far left resistor on your board) = 56K.
So they appear to be going a little longer than default delay.
It's difficult to calculate the actual turn on time since this board is operated at voltages lower than the recommended operating supply voltage (25V).
Usually this is an RC time constant for determining turn on time. Increase the value of the capacitor C2 to increase the time delay - maybe to a 56uF or 68uF and see what happens (56 & 68 are not popular and tend to be harder values to find).
You want a decent cap there. Any brand electrolytic will decrease in value as they age. As board gets older, the pop will gradually return until that cap is changed -- but this could take several years to reappear. Interestingly, they used three electrolytics on this board -- two piece of crap Lelons and one Nichicon UVR cap (decent cap). They used the better grade of Nichicon for the time constant. A Nichicon UVX type cap may be a better choice -- UVX is a higher temp rated cap which tends to have better seals/electrolyte and may last longer.