Marriage Sep quarter 1873 at Boston (ref 7a 693), between James Sykes and Frances Tilson, so it looks as though Nellie was living with her aunt and uncle. Doesn't explain where her parents were though.
Interestingly given that Nellie was Nellie Mason Hardwick on her birth certificate, there is a birth and death of an Ada Mason Hardwick as follows:
Birth Dec 1886 Spalding
Death Mar 1887 Bourn
Perhaps a sister of Nellie.
Do you have proof that Robert was born in Boston, his birth certificate for instance, stating he was from Boston on a marriage cert, doesn't mean he was born there. The reason I ask is that there is a Robert Hardwick on the 1901 census, who was born in Holbeach about 1852 living in Grimsby with a wife Betty S Hardwick. His occupation is given as Pork Pie Maker working from home on his own account.
Regarding the Mason name on Nellie's birth cert. I can't find any births for Frances or Kate Tillson (or Tilson), but there are births in 1856 and 1860 respectively in Lincolnshire for Frances and Kate Mason.
Still not found another marriage or a death for Kate.
Lizzie