KitKat,
I don't think they could class you as medically unfit because of a history of depression (they could only do so if you currently suffer from it). That said, I think emotional stability is one of the core competencies they ask you about at interview, so this may come up then and I can see how they may possibly decide a candidate may be unsuitable on this basis... it's a tough one, I guess you just have to be honest about it (both with NATS at interview and also with yourself when deciding whether or not this really is the right job for you).
Personally I have been a little worried about this myself, I also have a history of depression and am certainly going to be preparing myself to answer questions on it. In my case the depression was brought on as a result of an adverse reaction to medication I was prescribed for another condition, so I think I can reasonably argue that a) it is well in my past (no symptoms for over three and a half years now) and b) there is no significant evidence that I show any propensity towards emotional instability as the depression was the side effect of the medication I was taking rather than part of any kind of breakdown or failure to cope under pressure.