In the dark, run the engine.
Switch on the headlights main beam, and the heater fan and the rear window.
Rev the engine.
If the lights get brighter when you rev, then dim when the revs drop below
1000, the alternator is charging.
If there is no difference in the lights, the alternator is not charging.

Lock

The obvious possibility is that the brushes were almost worn out and
have no worn out. The dust in a panel shop wouldn't help matters.

Which does rather strongly suggest the problem is not the alternator.

Just to be sure to be sure.

Time to check the wiring.

Assuming it's a conventional internally regulated alternator, there
should be two main terminals on it.

One goes to the battery, typically with a heavy brown wire via the
starter solenoid on a Mini. That's just the output.

The other goes to the warning lamp, and from there to the ignition
switch and then to the battery. This wire connects to the field winding
input on the alternator.

When the ignition switch is off and the motor is stopped, both sides of
the warning lamp are at earth potential. The switch side is earthed
through all the other items which are on the ignition circuit, since the
circuit is off. The field winding side is earther through the field
winding itself and then the regulator, which is shorted to earth because
the battery voltage is under 14V.

When the ignition switch is on and the motor is not turning, the switch
side of the globe has live. The other end is still connected to earth
via the field winding and regulator. Current flows from the ignition
switch through the globe to the alternator and the globe lights. *If the
globe is dead* no current flows through it, through the field winding,
and so when the motor is started the alternator never starts producing
any power, so it does not self energise. The light does not come on and
the alternator does not charge.

When the ignition switch is on and the motor is running, the alternator
supplies the filed winding internally and that side of the globe is also
at 12V, so the globe does not light.

For the short time after the ignition switch is turned off before the
motor stops turning, the ignition switch end of the globe is once again
earthed throught the various other loads. The alternator end is live
from the alternator output, so current flows from the alternator through
the globe to the ignition switch loads and to earth and the globe lights.

Things to check:
Measure the voltage at the battery with the motor running. If the light
comes on only sme of the time, check with it on and with it off.

Check the field winding supply from the warning lamp is live when it is
 disconnected from the alternator and the ignition is switched on. If
you then earth it, the globe should come on. This tests both wiring and
globe. If they're OK, it's pretty much got to be an alternator or
regulator problem.

Finally, one more complication. Some alternators have a third, thinner
wire which ALSO goes to the battery + terminal. In a Mini this might
also be connected to the solenoid. This wire carries almost no current
and is for sensing the voltage AT THE BATTERY.

If this wire is simply connected to the alternator output at the
alternator, then it will work, just as older alternators did, but it
will not allow for voltage drop in the wiring to the battery. If it
comes disconnected, then the regulator will stay shorted to earth,
driving the field winding hard, and making the alternator charge as hard
as it can, often well past 14V, and boiling batteries. The warning lamp
should still work as normal.

Hi,
Do you have a voltmeter?
If you do, connect the meter to the battery and have the meter on a range
higher than 12 volts, (possibly 30 volts is common on most meters) check the
voltage and then start the engine. If the voltage rises to around 14 or 15
volts then the alternator is charging. If it stays at 12, or whatever it was
reading with the engine off, then it is not charging.

keith