Sounds like you might have a bare wire under your dash or behind your instrument cluster that is shorting out causing your fuse to blow
turn on your headlights
First check your fuses if none are blown then check the switch.
The headlights could have bad bulbs or a bad fuse; the two sets of lights are on different circuits.
check the fuses - there is no fuse for the headlights.
Turn on headlights/parking lights and adjust dimmer control
If you turn up your interior dashboard light settings, then the radio display will be brighter too. The radio dims to the setting for the dashboard when the lights are turned on.
Because the power for your headlights is pulled from your battery. When the dashboard lights dim after turning on the headlights the dashboard lights aren't getting as much power as they were with the headlights off. With the motor running or off? The headlights draw a lot of current, but should not visibly affect other systems. There could be a problem with a worn out battery, an alternator that isn't charging correctly, or you have a bad earth between the battery and the chassis. Check the large cable connection between the battery (usually negative these days) and the bodywork, or between the engine block and the bodywork.
Because it is British.
its probably something to do with the wiring system
Check headlights and dashboard lights first. If no/minimal power for the lights then replace battery.
Fuse for parking lights probably blown.
Some cars have a knob for the headlights. If this knob is turned left or right when pulled out, it changes the brightness of the dashboard lights from off to full brightness.