Does this appliance break UK safety laws?
Superficially that looks like a 'standard' triac dimmer circuit, and the capacitor isn't going to see more than a few 10's of V at worst.
The live wiring, although exposed inside the case, is not exposed outside the case - so isn't any more unsafe than it would be if the circuit was built on a PCB.
The oxide on the grey-insulated wire is probably from an aggressive flux in the solder. Not ideal, not particularly dangerous.
The quality of soldering looks something like I did when I was 10 years old, but I guess you get what you pay for ...