Humid air wouldnt affect the wiring unless it was extremely old and cut to pieces. Wiring laying in a puddle could be a problem even if the insulation is not cut.
I used to do a lot of fault repairs on electric machines, and nuisance trips are THE hardest things to fix, especially from a distance.
But THE most likely from my experience, I'm hesitant to suggest as you say you have zero electrical knowledge.
Assuming one of those mcb's is the lights and the other is the power, and its ONLY the power that trips?
Then I suspect you have a loose neutral connection on the board. Heres the reasoning;
The fact the washing machine trips it shows it isnt the lathe. The fact the lathe and or washing machine doesnt trip it every time shows its not a fault with either.
What happens is the loose wire sometimes makes or breaks the connection. When the current surges as from turning on or up, and the wire happens to be in the breeze at that time, there is a rapid pulse spark, just like a car spark plug. That spark will trip the mcb. Next time, the wire is laying firm against the connection and its 100% fine.
If you can FULLY isolate that unit from the house electrics, then pull the front panel and inspect ALL the wires into everything. Tug them quite hard. Any thing that looks lop sided or worn or burnt or loose, repair it and tighten down with a fair amount of force (by hand, no power drills).
If thats all ok, move along to the 13 amp wall sockets and again, tug every wire including the earths.
If on the other hand its the double isolator that trips, then you have to add in the lighting circuit. Does it ever trip when the light switch is operated?
As always, other options are available and it could be both machines, but thats where I would start looking.