As I understand this, you have two possible ways to go, Chamois.
One is to simply have the trunking removed, and the cables channelled into the wall, then plastered over and decorated.
Initially messy, but it's something that's done all the time. Any builder/handyman could manage this. Since the connections are not interfered with, it wouldn't need an electrician.
The other option is to call in a proper Heating Engineer, one who can handle the electrical work as well.
I'm assuming that the thermostat is a simple electro-mechanical type (the "spring" is just a switch that turns on or off depending on the temperature..... but this would have to be verified.)
Any simple thermostatic switch can easily be replaced by a normal wireless type. The trick would be to fit a "receiver" at the fireplace end, with whatever is needed to operate the controls.
Don't call in a plumber. You need someone used to complete heating installations, to figure out how it's all set up.