What are the voltages of these 3 wires relative to ground (black on AT connector)?

I do not know as I can not meassure them (start the PSU), but I do know what does it say on the PSU PCB Silkscreen:

Black > Ground 100%
White > ON/OFF
Red > +5VS

White may be an active high soft on signal. Perhaps try closing white and red with a 1k resistor. That'd give you some insurance if white provides a ground path.

If you do not have 5V on red when plugged in and switch in on position then its broken...

I have measured it today and it does have 5V on that wire. Should I short Red and White or is maybe something inbetween them on the board it belonged to (does it make sense to do something like that)?

Found something.. Language seems to be Polish. Maybe some online translator can help.:
AcBel model: API-6117L - how to run without a computer?

After letting Google translate the Polsku on that page, it pretty much says what we did. As we don't know what the overall soft power on circuit is, it's safest to experiment using a resistor in the range of a few hundred ohms (I said 1k) when you connect the signal wire to ground or +5V. The resistor will limit max current and reduce the chance of damage to components in the latch circuit.

Aye aye capt'n

If I'm the captain, my immediate order is for you to organize a mutiny. 😀

Yarr!
Because I couldn't find an 1k ohm resistor I tried shorting the 5VS Red wire pin and ON/OFF White wire pin with an 37k ohm resistor. Nothing happened.

Stojke wrote:

Yarr!
Because I couldn't find an 1k ohm resistor I tried shorting the 5VS Red wire pin and ON/OFF White wire pin with an 37k ohm resistor. Nothing happened.

If you have a multimeter, you can also confirm the loop impedance by measuring in circuit current (or voltage drop across your resistor) and therefore predict whether the current would be safe if you were to short the wires.

If this all fails, you're likely pulling open the PSU to a) understand the latch circuit being used, and b) find out what's broken.

Just to clear out any confusion,
the red has +5V standby voltage, the black is 0V ground and the white has what voltage (when standby supply is on)?

If white is near +5V, it means it should be pulled towards ground, usually below 0.8V to turn on, like with ATX supplies.
If white is near 0V, it means it should be pulled towards +5V, usually above 2.0V to turn on.

37k resistor is too high a value to turn it on, if for example there power supply has 4k7 or 10k pull-up resistor to 5VSB.

You should measure resistance between white and red to see what value of pull-up resistor it has.
You should also measure resistance between white and black to see if it has a pull-down resistor.
Do these measurements both when standby supply is on, and when the thing is totally disconnected from mains.

My gut feeling is that white floats at +5V through ~2k-10k resistance and you need to ground it to turn it on.

Note that some power supplies need a load or they won't start properly or at all, so add a one or two hard drives or car lamps there as a load, on both +12V and +5V lines. CD-Rom drives don't draw enough power so hard drives are better as dummy loads.