Not sure i get your question. If you mean the difference of it going through blank cables and through the saturation device then
this is why i posted the picture. It arrives practically in perfect sync. Not even a sample difference.
It is a maximum zoomed in screenshot of the initial Transient
In the picture the upper waveform is the blank cable and the one below is the signal through the hardware saturator (you can see the waveform is slightly bend harder-steeper cause of the saturation)
I cannot zoom any further but i see 0 timing delay difference in this resolution.
So i conclude, they arrive at the same time.
In case you mean something else please elaborate cause i did not get the question probably