I have an update. The 16t pinion arrived this morning, although it looks like I got the wrong pitch because it sounds kind of rough, so I've ordered another one which should be fine. In the meantime I've used this pinion to do some testing.
Here's a link to a video showing some testing.
http://www.sr8.co.uk/rc-car.wmv
What I've found is that if I put the receiver next to the motor at the rear of the car, where it's the only place it will fit, the car cuts out when I'm about 5-6 meters away. If I move the receiver to the front of the car, away from the motor, I can walk downstairs to the other side of the house with very minimal cut out.
During the testing I checked how hot the connectors and battery were. The connectors were fine, no heat at all. The motor was hot, but I could touch it for 3 seconds. The motor temperature is probably normal, if not, it could be down to the extra stress it's having to cope with while the wrong pitch pinion is on.
Because I wasn't at full throttle when any of the cut outs occured, then I'm assuming it's not a power issue, it's interference.
Will buying 2.4ghz radio gear definitely resolve this issue? I don't want to be wasting more money.
I forgot to mention, I've also put the capacitor on the receiver as suggested yesterday, they delivered pretty quick.