Hope the following helps.....................
Not knowing your car model intimately, or its year, I'll give you the Generic Low-down..................Like most early cars, esp English cars using "Prince of Darkness"equipment, aka Lucas, they rarely , if at all, used "relays". Consequently the whole circuit took the whole load till the weakest part failed. In your case, the headlamp switch has over-heated* and started to melt causing warpage and/or point burn-out in the HL sw, and so no high beam.
You'll need a new Headlight switch for starters, but check your globes and change them if they're anything but 60w/55w........ as often owners will put in higher wattage globes looking for better illumination (in reality doesn't improve much as the old wires and old connections can't handle the extra load)........If you want brighter lights and higher wattage globes, you MUST fit a relay for Low Beam plus a relay for High Beam, and ALWAYS have Fuses protecting ALL circuits, which is another cheap move by car makers, to leave so many circuits un protected and Fuse-less. This is a fire or accident waiting to happen. Try and avoid used HL switch; always new where poss.
Also check as many connections as possible. All would be ideal. Loose and semi-loose connections are the enemy of a good circuit. Pull them apart and clean them thoroughly with spray contact cleaner. Make sure ALL insulation of wires and terminals is sound.
Let me know how you go.
Bernie