Mine cut out the day randomly with a tacho straight to zero, popped the bonnet and headed straight for the wiring, the main ignition feed had come off the fuse box, no idea how, wasn't loose. Anyway, point is with your tacho heading straight to zero I'd concentrate on your wiring, and I'd head right to the beginning at the fuse box and work your way along to the solenoid and back up again looking for a loose/corroded connector or frayed shorting wire. It could be your electronic ignition but as you describe it reads like an instant loss of electrical power.
Wiring on a Dolomite is ridiculously simple, brown from the battery, White/Red to solenoid, White/yellow is the starter override to the coil when starting, white then Pink/Yellow (ballast wiring) to the coil. In the underwhelming fuse box green feeds an 'exciting' amount of kit, tacho along with just about everything else, hazards etc. Pink does lighting, horns and clock. White/slate is the tacho signal from the coil.
Coil does get hot if it is left on for even a relatively brief amount of time.
Current fleet: Triumph Dolomite Sprint '75, Daihatsu Fourtrak, Honda CG125, Yamaha Fazer 600, Shetland 570
Disposal fleet: Golf GTi 16v MK3 Anniversary
Past fleet: Triumph 2000, Lancia Beta Coupe, BL Mini Clubman, Austin Metro, Vauxhall Cavalier MK1 & MK2, Renault 18 D, Rover 216 GSI, Honda Accord (most expensive car purchase, hated it on pickup from dealer, was made out of magnetic metal as only car I've ever been crashed into, 4 times), Golf GTi MK3 16v x 3