Kev UK,
I looked at the 6066 site..some neat stuff. The diagram for the '63--'66 only shows the wire color/location at the connector on the steering column, not the switch functions.
If you assume the '63 switch contacts function similar to the earlier years ( a good assumption), then when you draw out what
doesn't work on your truck, because it follows no consistent pattern, it sure would indicate something wrong in the switch proper.
My only question would be the front turn signals. You indicate they work fine. Does this mean they are bright and flash at the proper speed.
Having a brake light illuminate while the key is off and TD handle flopped to one side would be proper...this circuit (brakes) should not be switched.
In the posts above, it appears that at different times, both brake lights will illuminate, so that means the wire integrity is OK. Sometimes bed grounding and lamp grounding can do strange things. When you do get the brake lights to illuminate, are they bright. Do the taillights function fine!
The easiest troubleshoot would be to pull the wheel and see if a handful of little brass contacts drop out in your lap. The 6066 guy first says only GMC parts will fit, but later I thought he alluded to the fact Chevrolet parts might work.
If you pull the wheel and expose the turn signal assembly, you might get lucky and find a paper clip that fell into the works...good luck.
Stuart