Owning your own home

Spot on the ceiling is quite odd. What's above there?

Issue is likely to be how invasive an "expert" is going to need to be to provide reasonably conclusive answers.

There are a limited number of possible explanations:
Rain getting in
Pipe leaking
Condensation

As it's a first floor flat, I guess we can eliminate ground moisture ingress.

I had this in my old flat, created by weathered (cheap) external rendering, thus making is permeable

It was difficult to diagnose, they spend a while inspecting the flat roof and gutters. It was probably a multitude of problems but they identified this as the main cause for ingress

My Building Management Co kept fobbing me off for months ( about 18 actually ) and it was only when I got the ombudsman involved that they finally fixed it. The longer they left it the worse it got - remind them that their neglect will end up more costly for them.

Sorry only just saw this.

If that patch of damp on the wall is a party wall it's likely that something's tracking internally from somewhere above.

The damp adjacent to the door could be the result of multiple issues but would be slightly more likely to be some kind of detailing failure on the external wall somewhere.

Usually your lease will state that the extent of your demise extends up to the inner face of the external walls. I'd bounce it back on that basis and try and be as forceful as possible about it..