Just checked again. He is named as Frank rather than Francis in newspaper reports.(and he was an electrician not a doctor) and was aged 37 in 1935.(oops, I see this information was given on your other thread).
This report in the Bexhill on Sea Observer was a tiny article on page 22. It may have been front page scandal in another paper.

I am thinking that of your grandmother was not legally married to Mr Walton, as he was already married to someone else, then your father would automatically have been registered in your grandmother's maiden surname.
You have stated that your grandmother "married" (in inverted commas) FEW. Does this mean that she lived with him but you suspect they did not marry? If FEW are his initials, are you able to tell us his name, or is he still living?