A fly off handbrake doesn't ratchet into position. Instead it acts like a brake for the rear wheels that doesn't lock unless you press the button. To release, simply lift the handbrake lever. The very last MG to feature a Fly Off Handbrake as standard equipment was the MGA. Some people believe that all true MG's must have a fly off handbrake. Others believe that there is a real advantage to having the handbrake work this way. If you are rallying or autocrossing this may well be true. It may also be the case that having the handbrake work in this way could deter theft of your car since any potential thief would press on the "Release" button expecting the brake to release and instead discovering that the handbrake is "jammed". Whatever the reason, if you wish to convert your handbrake to a fly-off type, here is how to do it. Before you read further, you may want to read up on how to remove and reinstall the handbrake lever here
Basically it involves making a new handbrake pawl, bending the pushrod the other way, and cutting the handbrake lever so it clears the pushrod. The new pawl essentially moves the ratchet point 90º counter clockwise. Be sure to leave clearance so it doesn't engage unless you press the button. You may wish to experiment with some shapes cut out of plastic or wood before moving to steel. Knowing this you can design your own pawl or you can use the one I have traced out below for the MG Midget. I know for a fact that this works for a 1970-1971 midget (at least). According to the Moss Europe Ltd catalogue, this handbrake was in use on all midgets and sprites with the only change being from chrome to black after car 160000 or so.
First, either obtain a Spridget handbrake from a junkyard or remove yours. Once out of the car take note of how it works. You have a button, a pushrod, and a pawl. Make sure that everything works smoothly before you continue.
Now it's time to disassemble the unit. First remove the split pins holding the pawl in place. Depress the button and slide it off the shaft and then the pushrod. With the pawl removed, pull on the button. You will notice that it is connected to the pushrod with a pin. Get a punch the same size as the pin, put the button in a vice and carefully tap out the pin. DON'T LOSE THE PIN!! You probably will not be able to replace it with a new one. There will be a spring inside, don't lose that either.
Now it's time to make a new pawl. I traced the one I made onto 1/4" graph paper so each square is 1/4" by 1/4" The actual image size is 1.803" by 1.507" at a resolution of 150 pixels per inch. Make sure that when you print it out at the proper size and resolution.
Transfer this pattern to a piece of steel that is the same thickness as the original pawl. I measured mine at about 0.185" thick. It is easiest to rough out the shape with a hacksaw and then grind down to the line with a grinding wheel. The holes need to be drilled just big enough to allow a good fit with about the same tolerance as original. Find drill bits that fit the original pawl's holes perfectly and use those to drill corresponding holes in the new pawl.
Next it's time to bend the pushrod. The idea is to bend it so that It works with the new pawl. This is easy enough to do in a vice. Now put the pushrod and spring back in and reconnect the button to the pushrod with the pin. Connect the pushrod to the new pawl and install the pawl on the handbrake lever. Mark where the pushrod bumps into the handbrake lever as it tries to retract the pawl. Finally, using a dremel or hacksaw and some files cut out some clearance in the handbrake lever so that the pushrod has room to fully retract. Make sure that it works and that the tooth on the pawl smoothly engages and is the right shape. test the handbrake by pressing the button and pushing down on the lever. If it stays engaged as long as you press down, you now have a fly-off handbrake. If not, reshape the tooth on the pawl a little until it does Put it back in the car and try it out. When installing frequently remind yourself that this handbrake doesn't release at the push of a button. I forgot a few times and laughed at myself for pushing the button and expecting it to release.