Further to info David gave...father Herbert Thomas Cole in 1901 is shown to be born in Thaxted Essex. So...here he is in 1881.It seems Herbert Thomas Cole's father is not another Herbert but Joshua George Cole.
Living at Tannery, Great Dunmow Essex
Joshua Geo. COLE Head 41 Thaxted, Essex, Builder Valuer Employing 4 Men And 1 Boy
Mary Ann COLE Wife 38 Colchester,Essex Builders Wife
Henry Geo. COLE Son 15 Thaxted,Essex Builders Son
Agnes Mary COLE Daur 12 Thaxted,Essex Scholar
Herbert Thos.COLE Son 10 Thaxted,Essex Scholar
Clara Annie COLE Daur 8 Thaxted,Essex Scholar
Emily Bertha COLE Daur 6 Gt Dunmow, Essex Scholar
Florence Julia COLE Daur 4 Gt Dunmow,Essex Scholar
Elizabeth Rose COLE Daur 2 Gt Dunmow, Essex
Mercy Jane COLE Daur 11 w Gt Dunmow, Essex
Laura Louisa BRADBROOK Servant Springfield Essex

The family has moved to Bedfordshire by 1891-Thomas's parents die in Bedfordshire as shown on Free BMD
Deaths:
Joshua G. Cole Dec 1915 Bedford Reg. Vol 3b pg 385(age 76)
Mary A. Cole Mar 1898 Bedford Reg. Vol 3b pg 224 (age 55)
Thomas's parents marriage:
Mar qtr 1864 Colchester Reg.
Joshua G. Cole
Mary Ann Seabrook
So the family appears to be an Essex family originally,that move to Bedfordshire before 1891.
Interestingly I found Joshua & Mary mentioned on Essex History House site
http://www.historyhouse.co.uk/placeT/essext04v.html
Case
Defendent:Thomas Merritt 26 stone mason
Victim's:Mary Ann Cole,John William Cole,Joshua George Cole
Fraud
Thaxted
Calendars of prisoners for Essex Quarter Sessions, Essex Assizes and Special Sessions of Gaol Delivery,
1866 Q/SMc 9, Essex Record Office

Hope this helps
Pamela