Hi, I too own a Mini Countryman with the N47 1.6 Diesel engine in. Ours has 15,000 miles on the clock, its made this noise since we had it at around 7000 miles. Here is a video, hopefully you have the same noise and i'll finally have found someone that has the same problem! It's been in to the dealers numerous times but they just say its a characteristic of the injectors under load. Thing is, it doesn't do it all the time so it can't be when they are under load because surely it would do it every time?
Now, the warranty is due to run out in October so I've booked it in again and pushed to get a drive of another similar age Countryman Cooper D to see if the same noise occurs. I think it makes the engine sound so un-refined and much like a very old Land Rover or early 90's Peugeot diesel, I'm sure I've not heard any other Cooper D making this noise when I've seen them around... It only happens at low revs, say when pulling away from stand still in first - also if you're creeping along in second and then ask for more power, it does it then too and gradually disappears as the revs get higher. Hope someone can help, or at least we can identify if we share the same problem!