Professor Bevan, of Aberdeen Royal Infirmary, spotted that Sister Aelred, of the Arundel Convent in West Sussex, was speaking as though her tongue was too big for her mouth.
The senior consultant endocrinologist believed the nun was suffering from acromegaly, a condition that makes the body produce too much growth hormone, and can shorten life by up to 10 years.
Mr Bevan said: "I was watching because Whoopi Goldberg was on speaking to these nuns trying to plug the Sister Act stage show. When the video greeting came on, I said to my wife, 'The middle nun is speaking as though her tongue is too large for her mouth.'
"I noticed she had thickening of the soft tissue of her face - the lips, the nose, the forehead - all of which are fairly classical external signs of this condition."
After the The One Show programme had finished - aired in 2010 - Mr Bevan tracked down the producer who contacted Sister Aelred. Following Mr Bevan's advice, the nun saw a doctor and the diagnosis proved correct.
Acromegaly mostly affects adults in middle age and can result in severe disfigurement, complicating conditions and premature death if unnoticed.
Sister Aelred has since had surgery on her pituitary gland to remove a benign tumour that was causing the condition. She recently met with Professor Bevan and his wife at the Arundel Convent to thank him personally.
Sister Aelred said: "It seemed extraordinary that a consultant whose speciality was endocrinology should have been watching and should have noticed and taken the trouble [to contact the producers]. I'm very thankful for Professor Bevan's help.
"When I met him, I told him I would be forever in his debt for taking the time to diagnose me by watching me on television.
"I didn't believe him at first, but when I looked up the symptoms I realised he was right. I didn't feel ill at the time and would have gone on until it was too late.
"Acromegaly has a number of serious consequences. It could have locked my jaw shut so I couldn't eat; it could have damaged my internal organs; and apparently it can make you more susceptible to diabetes and cancer.
"I underwent surgery in April 2011, but it wasn't completely successful, so I've also had radiotherapy. It takes a few years to work, but so far it seems to be working and within a few years I'll hopefully be OK again."
Mr Bevan said he was glad to help Sister Aelred as her condition is one that can be incredibly hard to spot. He said: "Acromegaly often happens very slowly and often happens in the second half of life. It is very difficult for the individual themselves to pick up.
"In every one million of the population there will be probably three or four new cases of acromegaly diagnosed, so it is very rare."
A spokeswoman for NHS Grampian said: "Professor Bevan is very modest about what he has done for Sister Aelred, but he has given her a much longer life."