Your indicator is blinking really fast because one of the bulbs on that side is blown - put your hazards on and check all bulbs around the car, if any don't work you will fail MoT.
On the side of the gearbox somewhere is a switch that is thrown when you select reverse gear that causes the reversing lights to illuminate. You can test this circuit by unplugging the lead that runs to it and shorting it with the ignition switched on. You will not fail MoT if the lights do not work.
Whilst you are there, ask an assistant to press the brake pedal and check all of your brake lights, including with your headlights switched on. If one side illuminates only, and at the same time the other side's tail light goes out, then your light unit on that side is not earthed properly - this is a common fault. Undo the earth wire where it bolts to the car body and clean it up with some emery paper. You will fail MoT if any lights except reversing lights do not work.