Does it crackle for a couple of seconds and then quit? If so, it's your fuel pump. Both of my Triumphs and my VanVan do it, but not every time it switch the ignition.
Now, after the broken oxygen sensor was replaced, it happens rarely but it can be a couple second, or 5, or a couple seconds then silence then a couple seconds again. The pattern is irregular. What you can hear on SoundCloud was recorded before new oxygen sensor was installed. I'd say it rarely happens nowadays.
Provided it's a fuel pump what is the connection between these sounds and the oxygen sensor? Also, what does this sound mean? I.e. what is it that the fuel pump does that produces this noise?
I assume the fuel pump would be some sort of piston system. That's what mine sounds like. I have no idea in what way that might be connected to the O2 sensor. It seems odd that they would be related at all.