I'm assuming you don't have the marriage certificate of Thomas Peterson and Lily Hart who married 1930 as it would appear you have definitely been following the wrong family.
Your grandmother was plain Lily as shown at birth, marriage, 1939 register and death and, as you say, she was indeed born 7/7/1908 - as already posted, her birth entry was Sept.qtr.1908, mmn Watkinson and, again as already posted, her parents were Thomas Hart and Mary Watkinson married 1886. I can find at least 11 children born to them:
Mary - born and died June quarter 1886
Edith - birth June quarter 1887
Thomas - birth and death March quarter 1892
Thomas Henry - birth March quarter 1893
John Harold (Harold on census) - birth June quarter 1895
Mary Alice - birth Dec.qtr 1897
Elizabeth - birth Dec.qtr.1898
Eva - birth Dec.qtr.1901
Robert - birth Mar.qtr.1904, died June quarter 1905
William - birth Sept.qtr.1906
Lily - birth Sept.qtr.1908
(All the above registered at Sculcoates, Yorkshire)
So, your nanna was Lily Hart born 7/7/1908, dau. of Thomas Hart and Mary Watkinson (and not Lily Marjorie Hart born 8/11/1907, dau. of George William Hart and Lily Westmorland).
Census confirms that Thomas, Mary and all their children were born in Hull.
Annette