Bzz1,
Would you rather jump through every hoop and get through Stage 3 only to find, when you get to the medical, that NATS won't employ you, or find out now without having to go through everything first? I know which I would prefer... perhaps there isn't an advantage writing 'unknown'.
I don't think NATS will 'automatically' reject you - most people get sent for an eyesight assessment. As for RN/RAF, you are a officer/soldier first and an air traffic controller second, so there are differences there, and also, they don't expect you to work for them for the next 30+ years.
Good luck with HR/your optometrist.