How do I cut a channel and plaster television cable work in a brick chimney?
The brickwork on a chimney breast of a 1900s house will be solid brick (not veneer).

Cutting a channel, fitting trunking and replastering is eminently sensible if that's the only way to go. It's also a job you can do your self with a masonry drill and cold chisel. Using an SDS drill with a chisel attachment will be quicker if messier.

Make sure you take the cable vertically (if possible) never diagonally across the wall. This will make it safer for future owners of your house.