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Through the loft it will be bare wires, but instead of just having wires dangle from the ceiling I think tucking them into a conduit of some kind will be tidier.
The rooms are 15m apart, and there are 3 brick walls, 1 chimney, and 4 doorways in between. Hence... just go over it all.
The loft is very accessible, so the only thing I really have to do is drill a few holes in the right places.
Conduit isn't necessary in the loft, although it may be a bit tidier (depending on whether you have flooring up there, and which way the joists are running).
What are your walls covered in - It may be neater to go behind it they're stud / plasterboard.
Routing from top down is also easier than bottom up.
Also, with cat6, you don't need to run the shortest route - signal is good up to 100m, iirc. Meaning you can route wherever keeps it out of the way.