Ken57 explained :
From inside, the roof seems to be made up of concrete beams arranged
like floorboards, with joins in places that make me wonder what
they're attached to.

From on top, it looks like crumbly concrete covered in various layers
of patching and gunge, (some of which I've just peeled off as it was
detaching itself), with several breaks in the level.

That looks very like concrete poroured in situ, which was supported by
floorboards whilst it was allowed to cure. The mess on top looks as if
it was once covered in pitch to provide waterproofing to the concrete,
later patched with the grey material.

If the concrete looks serviceable, then just get a flat roofing
specialist to take a look at it.

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Regards,
Harry (M1BYT) (L)
http://www.ukradioamateur.org