If it was a small business or a charity then I think there would be a moral dilemma, but if the job isn't with either then go for it. If an employer really wants you then they will invest in you and will be patient.
I think the real question is at what point during the process you mention it. Whilst it shouldn't matter, the truth is that there is always a risk that disclosing it early on might put the employer off (even if subconsciously). My hunch is to not mention it until they offer you the position, but it's probably best to speak to a recruitment consultant or careers adviser about it.
Anyways, good luck with it and get well soon.
It is a large organisation, but my main boss has his fingers in many pies and will find out I've applied.