This is a clever circuit and practical as well.
However it puts low-power circuitry in close contact with high power...
In such a case you need to ensure cooperation between the two.
This circuit should be tested in sections. The timer circuit should be tested for proper operation without being hooked up to the welding circuit.
Then take the next step. Wrap a few turns of coil L around the welding cable. Notice that the coil L is not directly connected to the welding cable. It's only wrapped around it.
Wrap just enough turns to result in a usable signal to pins 2 and 6 of the 555 IC.
Do some welding. Stop. See if the machine turns off at the end of the timed delay.
I have doubts about the wire containing switch K2. It connects delicate components to the welding cable. When welding stops, the transformer coil may generate a hi-voltage spike. This may destroy the timer devices. Maybe it happened already.
I suggest adding a resistor in that wire between welding cable and K2. Start with a high ohm. Reduce resistance until you get proper operation.