As I recall, the answer to the question in my headline is "almost certainly not". But the programme does suggest that the Mitford family may have exaggerated how ill Unity was after her suicide attempt so that they could brush off awkward questions when she returned to England.
But Martin Bright's New Statesman article on the story from last year is funny and informative:
Val was a little nervous as she explained that her aunt Betty Norton had run a maternity home to the gentry in Oxfordshire during the war and that Unity Mitford had been one of her clients. Her aunt's business, in the tiny village of Wigginton, had depended on discretion and she had told no one except her sister that Unity had had a baby. Her sister had passed the story on to her daughter Val.Incidentally, Unity Mitford was the granddaughter of Thomas Gibson Bowles. This blog isn't just thrown together, you know.
I casually asked who she thought the father might be and there was a short silence on the other end of the line before she said: "Well, she always said it was Hitler's."
I must say I was tempted at this point to put down the phone. Christmas was coming and I was very busy. But for some reason I decided to carry on listening to this bizarre tale. Val didn't sound mad, and she said she was merely passing on a family story.
The child was a boy, she believed, and he had been given up for adoption. She didn't want any money; she just wanted me to look into it. So here was the prospect of Adolf Hitler's love child alive somewhere in Britain - it was either the scoop of the century or completely bonkers. But it had to be worth a few hours of my time, even if it turned out to be a dead end.