Thank you for your reply. I was very interested to read it, particularly as the dates fit and Pinder is a rare surname. However, having done some research today I have to conclude that we are talking about two different Catherines.
The GRO index, 4th quarter of 1848, has the birth of a Catherine Pinder in the Islington registration district (the volume is III and the page 265), which is probably yours, although it is London not Birmingham. My Catherine was born in Belper according to the 1851 and 1861 censuses, and there is a birth recorded in the 3rd quarter of 1847 which is probably her. This would fit with the ages given (3 and 13) in the censuses - someone born in Dec 1848 would be 2 in 1851.
I also checked the 1881 census and couldn't find my Catherine (I suspect she had either married or died by then), but there is a Catherine Pinder age 32 living in London and born in Islington. She is said to be the niece of John King (the head of the household). Does this sound like your Catherine?
If you would like the full 1881 entry please e mail me.