Susan,
As you say censuses 1851, 61 and 1871 have William as born Thorp/e.
1881 census has him as just born Tudhoe, where he was living in all censuses from 1841 (304/19 3-1) to 1881. So in at least for 40 yrs of his life he did not move from Tudhoe- where he was born ( according to 1881 census)
From the Enumerator's discription of the Enumeration District in the 1841, the part of Tudhoe he lived in/at, lay within the Parish of Whitworth ( other parts of Tudhoe lay within the Parish of Brancepeth)
So although I cannot find a Thorp/e in the area of Tudhoe on old maps/documents, I think Thorp/e was in Tudhoe vicinity.
Trivia Angle. The word " thorp" or "thorpe" is Old English for " small village"
In the database of places covered by Registration Districts from 1837, there were over 50 places in England called Thorp or Thorpe with or without suffixes.
Michael