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UK home, wiring the chimney?
(alien.top)
Is the chimney capped at the top? If so, and there’s not a lot of moisture ingress, you should be fine. I’d run some fibre alongside the Ethernet too so you can run 10GbE or 25GbE in the future if you wish.
If they're Cat5/6 cables, in the U.S., they're low voltage and don't need a licensed electrician.
You're supposed to support the cables by attaching them to the surface. You're not supposed to leave them dangling or lying. They don't require a conduit though. If you get quality outdoor Cat6 cables, they're probably well protected in a dry covered chimney.