I now believe the Joseph Fletcher, witness at Ann Jenkins Fletcher's marriage, was her brother and not her father.
1841 Census - Dudley, Worcs. (reference given by Keyboard86)
Ann Bridgwater 20 Ind. Not born in county
Ann Bridgwater 1 mos. Born in county
Charles Fletcher 20 Cl. Not born in county
Joseph Fletcher 20 Pattern Maker Not born in county
Mary Fletcher 5mos.
Nancy Fletcher 20 Born in county
Frederic Fletcher Not born in county
+ a servant
So, children of Joseph Fletcher and Mary Jenks were:
Joseph Fletcher bc.1816/7 (cannot find baptism but in 1851 states born Little Wenlock)
William Fletcher b.early 1818 (aged 2 yrs 8 mos at baptism 20/10/1820 Broseley)
Ann Jenks Fletcher b.1820, bp.20/10/1820 Broseley
Charles Paul Fletcher bp.20/10/1822 Broseley
Harriot Fletcher bp.24/9/1826 Westbury
Mary Jenks Fletcher bp.23/3/1829 Worthin
Frederic Fletcher bc.1831/2 presumably in Shropshire too
Joseph Fletcher Pattern Maker, son of Joseph Fletcher, Chemist, married Nancy Bridgewater, dau. of Joseph Bridgewater, Grocer 17/12/1839 West Bromwich - Witnesses William Fletcher and Harriot Fletcher (his brother and sister) i.e. Nancy Bridgewater was the sister of Ann Jenks Fletcher's husband Joseph Bridgewater. The Mary Fletcher 5mos on 1841 census Dudley was the daughter of Joseph and Nancy and the Charles 20 - shown as Clerk - is your Charles Paul Fletcher, and the Frederic 9 another sibling.
So, where were William Fletcher bc.1818, Harriot Fletcher b.1826 and sister Mary Jenks Fletcher born 1829 in 1841 census? Two siblings married in West Bromwich, Staffs. but were in Dudley, Worcs. in 1841, 2 siblings (CP and H) married in Birmingham but by 1851 Harriet and husband have moved to Stoke upon Trent, Staffs. and Knutton, Staffordshire in 1861.
The only sibling that seems to have remained in Birmingham was your Charles so don't think the family were ever there 'en masse' and that Charles moved there sometime in 1840's together with sister Harriet.
Annette