Origins of the phrase 'titty flo'
Solution 1:
Perhaps surprisingly, the first definition of titty in OED does not refer directly to breasts.
titty n.1
Sc. colloq. A sister; a young woman or girl. Cf. kitty n.1
While Oldham is hardly Scottish, it's not beyond the bounds of possibility either for a dialectal word to have worked its way south, or for OED not to have recorded other relevant regional usage. The OED entry has citations between 1725 and 1896, but it hasn't been updated since 1912 so there isn't any recent usage data. It may well have been around in our grandparents' youth, even if it's not used much in that sense nowadays.
As for Flo, it's a female name (short for Florence), and sounds good paired with titty. Much as Andy Capp's wife is Flo, I suspect it's a generic regional name.
— Andy Capp by Reginald Smythe (Mirror Group Newspapers) via cartoons-comics.deepthi
Solution 2:
The following information has come to me from friends on another website, so I can take no credit for it.
It appears that Titty used to be a diminutive of Letitia (or Laetitia), and Flo, of course, is a diminutive of Florence, as has already been pointed out. We may, therefore, be looking for someone named Letitia Florence, and there was indeed a Laetitia Florence Findlay (1903-1978), the daughter of a Scottish baronet.
In 1925, Laetitia married Sir Hugh Vere Huntly Duff Munro-Lucas-Tooth (1903-1985), 1st baronet, to become Lady Munro-Lucas-Tooth. The following photograph, taken in 1926, shows Sir Hugh and Lady Munro-Lucas-Tooth, with the baronet attired in court dress.
National Portrait Gallery, London
They had three children, the eldest of whom, a daughter, was also named Laetitia (but not Laetitia Florence).
Sir Hugh was M.P. for the Isle of Ely (Cambridgeshire) between 1924 and 1929, was called to the bar in 1933, and was M.P. for Hendon South (London) between 1945 and 1970.
Laetitia became a Justice of the Peace (magistrate) in 1951 and was awarded the O.B.E. (Order of the British Empire) in 1958.
It is possible that Laetitia became known to the general public because of her husband's political career, but finding out if she was known as "Titty Flo" may be next to impossible, given the unfortunate association, and the vagaries of Internet search engines.
Was Lady Munro-Lucas-Tooth your grandmother's "Titty Flo"? It would be nice to think so, but unfortunately, it is likely that only your grandmother could tell us.