If the 1901 birth year is correct, there don't appear to be any William Smiths in the 1901 census, with father Alexander, born 2 years +/- 1900 apart from the Jarrow one.
It is tempting to think he is the correct one as he is the only one.
There are, however a few in Scotland ....
Added: I'm not sure how accurate the interpretation of an accent by a 10 year old would be, especially 'X' number of years later ... I would also be asking the Great Grandaughter where she heard that William was adopted.
