I'm trying to recall this long article someone wrote. Her mother had moved her and her family to Canada for witness protection. It turned out though, the mother was under the influence of someone who was deluded into thinking they were out to get her. It took years before the author came across evidence to the contrary. Her father and her aunts were not dead after all. At which point, the person who was deluded said, oh, that's cause they had been replaced by clones.
I make a distinction here in that, I think the mother here was herself not delusional. She was influenced, and perhaps, could herself reason herself away from it given time. The person who was delusional, however, will continue to generate new explanations to fit the evidence around that delusion ... and is probably not even aware of it. According to the author, he was otherwise rational in everything else but his delusion.
(apologies if that's one of the BBC's randomly unavailable-outside-the-UK pages...)
Isn't that the required quality for it to be a delusion? The moment you realise that a thought or a perception is a delusion, it is no longer one.
Also, this is the story you're thinking of: https://www.bbc.com/news/stories-42951788
What would you call persistent beliefs that you know are erroneous, and yet you cannot somehow shake?