If you need a wiring diagram for your car, copy the PDF linked below. The later model MGBs are covered on the last couple of pages.
http://www.advanceautowire.com/mgb.pdf
When trying to start the car do you have any status lights on the dashboard?
Which fan is making noise? Is it a shop fan or the cooling fans on the car?
The car's cooling fans should not run when the key is turned to run unless the temperature switch is closed. Since your engine isn't running (and therefore not hot) the cooling fans should not be running.
The following is just a quick overview of what should happen.
You turn the key to the START position. The white/red wire on the ignition switch should have 12V on it.
When the white/red wire has 12V on it, the starter relay should close with a soft click.
When the starter relay closes, the starter relay should supply 12V to the white/brown wire going to the starter motor solenoid.
When the white/brown wire has 12V on it, the solenoid on the starter motor should fire with a loud click.
Then the starter motor should spin the engine.
With the help of an assistant (and a multimeter) you can measure where and if you have 12V on the starter circuit. You'll want to measure between ground the the white/red wire and then ground and the white/brown wire.