Go to this site and download wiring diagram for your model year car. On the left click on stock schematics, then MGB.
http://www.advanceautowire.com/
Where is the jog switch connected? knowing that may help eliminate some checks. Your car should have a start relay. It's just behind the fuse box. The first check I would make is to have an assistant turn the ignition to start while you have your finger on the relay, if working you can feel and hear the relay clicking. If it clicks the relay coil should be good but the contacts inside the relay may be bad. If no click, turn the ignition to start and check for 12 volts on the white/red wire on the relay. If it doesn't have 12 volts trace the wiring back toward the ignition switch and repair as needed. If you have 12 volts get a jumper wire with two clips on it. Make sure the transmission is in neutral, hand brake on and your feet not in front or behind the wheel. Jumper the brown relay wire to to the white/brown relay wire, does the starter crank? If yes I would check or replace the the relay. The OE type relay is rather expensive, you can use standard relays from NAPA, Advance or others. The terminals will be labled differently. The white/red wire connects to relay terminal 85, black to 86, brown to 30 and white/brown to 87.
Clifton