I'd agree that tucking back out of the way is a recipie for trouble.
Some further thoughts based upon the new image:
There's a label on the inside of the thing which will tell you for sure, or at least will give a model number which you can probably stuff into google- This is just visible inside the lid on the left of the new picture.
I think there's also one on the bottom thingy, on the base near the earth wire.
Assuming it's a transformer, are (were at some point) there any downlights in the room directly below?
Don't assume bath lights immediately.
Absolutely spot on! And I didn't even notice. Much appreciated!

The kitchen below the bathroom - the wires coming out are the same as the ones upstairs.
and be extraordinarily careful that there's not a live wire in amongst it somewhere!
Well the lights in the kitchen still work so I'm slightly concerned my live wire detector off Ali Express might not quite be up to the job.
I think I'll leave well alone for now - I certainly don't like where they are but I'll have to think about it.
Many thanks all!