Two options:

 

Your flasher unit is crapping out or it's the wrong model, and has too much electrical resistance, requiring a higher voltage to do the flashes.

 

Your generator/alternator is crapping out and doesn't have enough power to run them both.

 

Had the same problem with the moke, the turn signals would get slower and slower until they would flash no more. Turned out to be the flasher unit.

Yep.  You are right Hugh.  Sorry.  I was thinking it was the brakes and turns.  My mistake.  

Regardless, all the bulb filaments work independently.  

No Doug, on Minis, the turn signal bulb and the brake light bulb are two separate bulbs. The turn signal bulb should be a single element bulb at the top and the dual element bulb is at the bottom (or middle, depending on the year) combined with the running/tail lights. Sounds like either the dual element bulb is being used as both the turn and brake light, or if you have the brake light switch on the pedal assembly, perhaps a short/grounding issue? Get a wiring diagram and check your connections.

The obvious first answer that is bound to be repeated in future posts is to check the ground connections of the lamp housing assemblies.  However, that is not likely to be the problem.

Look at the wires going to the lamps and compare their color codes to that shown in your car's wiring diagram.  I suspect you will find that someone has somehow connected both the brake lights and turn signal lights to the same wire... somehow... or there is some malady at the lamp sockets themselves.  Though brakes and turn signals use the same bulbs... they are dual element bulbs so one should not interact with the other.

When my brakes are applied my turn signals do not flash (either left or right). Once i let off the brakes the turn signal now works. It might be one of many things but what should I check first?.......Thank you