In the junction box...
I can see 6 wires.
3 on the grey cable (thicker)
Blue.. Neutral
Brown.. Live
Yell/green.. Earth
These are the live feed and I would expect these to connect to the plug.
Please check this cable going into the plug is grey?
3 from the smaller white cable (thinner wires)
Blue neutral.. Ignore
Brown.. Live to stat
Black.. Switch live from stat..
These are the feed for the light and switching. I would expect these to connect to the stat.
Please check this cable (white) is going into the fridge compartment.
To test the compressor, if I was fiddling with this. And I am not a Frige engineer or an electrician. I would make sure the plug lead is grey and unplugged.
I would the pull the black wire out and isolate it. I would then pull the brown out and put it where the black was.
I think that will bridge the switch, making it "on" as soon as you plug it in & flick the switch.
If it came on, (as opposed to making a fooking great bang and creating an unpleasant smell immediately behind you) I would leave it on for a bit to see what happens.