Is this his sister?
Name: Ellen Cartwright
Birth Date: 17 Jun 1827
Baptism Date: 31 May 1828
Baptism Place: Birmingham, St Philip, Warwickshire, England
Father: James Cartwright
Mother: Maria Cartwright
Reference Number: DRO 25
Archive Roll: M40
This could be him, although the mothers name say's Eliza the rest of the details match.
Name: John Cartwright
Gender: Male
Birth Date: 30 Jan 1824
Baptism Date: 1 Jan 1825
Baptism Place: St. Philip, Birmingham, Warwick, England
Father: James Cartwright
Mother: Eliza
FHL Film Number: 502185
Reference ID: yr 1825-1825 p 306