Seems to be quite a few records available for your Caldicotts.
Baptism for George 15 Feb 1840 St Phillip's, dob 19 Jan 1840, son of James Coldicott and Emma Coley.
James Caldicott married Emma Coley, 15 Jan 1829 Harborne.
Baptism for James 1 Jul 1805 St Phillip's, son of Robert and Ann Caldicut.
Ann b1771 is on the 1841 census living in Barr St with 2 sons, Edward and Samuel.
Neither of James and Emma's children born after 1837, George or Arthur, seem to have been registered. There is a George Caldicott registered 1839 but he appears to be the son of James' brother Benjamin.