Got the same problem on a Mini One 2005.
Apparently 2003 - 2006 models are effected. If you remove the airbox you can see the selector. The ball pulleys are probably free but the neutral is the block in the middle that should move freely up and down and you should be able to do it with your hand. If you can't it means that you have a problem with the selector. Replacement via Mini will cost in the order of £180 for the part plus fitting. Or.... you can do a repair - which I intend to do tomorrow. If it works - and I have no reason to doubt it - I will post again on the forum and let you all know how I did it and add some pictures.
Basically you have to remove the selector (three bolts) Support the gearbox from underneath (jack) and remove the selector. You cant do it without first removing the engine frame bracket.
There should be an obvious link that is seized. You grind off one end and put in a new link (Pic attached) which I have been told resolves the problem.