I'm guessing that the family has used NICHOLLS CROOKES as a double-barrelled name?! Otherwise the elder Ruth would not have it - but it's a handy marker while searching for these people. Nevertheless I still can't find them in the 1901 index.
I took a look for a marriage for the daughter and I think I got lucky:
GRO Marriages DEC Qtr 1921
BRADBURY Harry T
CROOKES Ruth N
Ecclesall Bierlow 9c 879
I didn't look any further, and this might not be the right one, but the following seems to add a little weight to it:
White's 1919/20 Directory of Sheffield
Pomona Street
45 Bradbury Harry chauffeur
51 Crookes Herbert Nichols (j) table blade grinder
j=journeyman
By this time the birth indexes record the mother's maiden name so you could search them for Bradbury children with Crookes as a maiden name starting with the date of the marriage and working forward.
My particular interest with these people is the following burial in Walkley Cemetery (I am the chaiman of the Friends of this cemetery):
[name/date of burial/address or place where death occurred/ age/grave]
CROOKES Herbert Nicholls 9 Aug 1926 51 Pomona St. 54 G37
There are two other burials recorded for this grave (so far: the database is not quite complete):
LINTON Pearson? 24 Dec 1930 86 Freedom Rd 80 G37
VOCE John 5 May 1931 12 Herries Rd 65 G37
Note: 12 Herries Road means the Hospital that later became the Northern General Hospital.
I have no idea what the connection might be between these people, but it appears from my plans that there is a gravestone, not yet recorded. There is no sign of the Bradburys or the elder Ruth in the cemetery.
Do you know where the name NICHOLLS comes from in this family?
Hugh in Sheffield