If there is no movement in the pedal then that means the discs are OK, they're not warped. The noise will be, as others have said, rust or worn brake pads. Seeing as the car has passed the MOT then that should mean the pads are also OK so that leaves rust. If it's not too bad, it can be removed by gentle braking.If that doesn't work then the discs need to be cleaned with a wire brush or a sanding machine. If they are very bad and the rust has pitted the discs then they need to be replaced.