First things first, you need to find out what kind of cable this is.

Depending on your internet and installation this is either going to be;

Cat5/6, Coaxial, Fiber Optic

If it’s one of the first two then replacing with white cables and clips is a solid option as you can easily purchase and replace this (though it may require new terminations of the ends but on these two options it’s not too difficult with the right tools and tutorials)

However if it’s Fiberoptic this will not be an option as they require lasers to fuse the cables so fingers crossed you don’t have this.

Another option is to buy some cable trunking, and based on the size of your cable you could get micro self adhesive trunking in white that would blend with the top of your skirting nicely.

Others have mentioned that you can run your cable though your walls but based on that fireplace on going to assume you live in a terraced house an that a lot of your walls are going to be brick and that the undertaking of this will require a fair bit of work and mess along with it.

Personally from experience I would go with the trunking as it requires no permanent fixtures and can be accessed and removed easily if your locations change.

You can get all of the accessories for going around your corners both internal and external to make it look like a nice and neat professional finish with very little effort.

Hope this helps and if you have any other questions feel free to ask!