Not sure about any connection in the past. They were both for sale in 1965 - a description at the time:
"The Avon water runs under Gordleton Mill which dates from the Queen Anne period but which has considerable Victorian extensions. There are six bedrooms, three bathrooms and a swimming pool and cottage in its eight acres of land. Around £25,000 is being asked.
The stream has been diverted from under Flexford Mill which is about as old. The house has six bedrooms, two bathrooms and two kitchens, as well as a library, study, drawing room and dining room. There are 18 acres of land. Asking price £27,500."
Sunday, July 25, 1965
Publication: The Sunday Times
You can see both mills on this 1896 map. I'm not an expert on Mills, but don't think they would have had any connection - except the river!
https://maps.nls.uk/geo/explore/side-by-side/#zoom=16.2&lat=50.76885&lon=-1.59425&layers=6&right=ESRIWorld