When you say that Fanny "took her mother's surname at birth" do you mean that Edwards was her mother's maiden name, or that perhaps her mother later married a Mr Edwards, and we'd need to look for a different name in the index?
KGarrad has found the only Fanny Edwards birth registered in the September Q of 1913, and there are two more registered in the December Q. All three were registered to married couples (or to couples claiming to be married). None in Lancashire.
If it is the case that Fanny's mother was not an Edwards, but later married a man of that name, there is a possible birth registration in Oldham district, September Q 1913 of Fanny Whitehead Mmn Whitehead. (If there was a father's name recorded at registration, this would show as a separate entry in the index with the same page and volume numbers. There is no such entry.)
Marriage at St Margaret, Hollinwood, Oldham 21 February 1914
Lily Whitehead 26 Spinster, winder of 48 Fox Street, Hollinwood d/o John Whitehead, brick setter
Joseph Edwards 34 Bachelor, cop packer of 49 Argus Street, Hollinwood s/o Samuel Edwards, collier (deceased)
witnesses: Leonard Taylor; Jessie Lang
A possible child born to this marriage is Jack registered Oldham district September Q 1914.
A tree on Ancestry shows that this person married Mary Wild in Oldham in 1938 and died in Brisbane, Queensland in 1996 (no parents or siblings shown).
