I've drawn a blank with Uriah too. I tried looking for his wife-to-be to see if he might turn up with her in 1871, a year before their marriage, but she is a difficult lady to track down too.
I think I've cracked the case for you with Uriah's sister however.
I looked at a scan of the 1851 census and to me her name looked like Leviniah. In the 1861 census return for Kings Sutton the writing is often difficult to decipher, the name looked like Anna with an H squeezed in front of it afterwards, but I could make it read Lavinia if I tried (I wasn't too worried about her age being a couple of years different from what would be expected, people's ages vary so much on census returns). No sign of Lavinia / Leviniah in 1871. But what if she had married by then?
I tried FreeBMD for marriages of Lavinia Co*ns (to cover all spellings) in Northants and Oxon between 1861 and 1871. Bingo - Lavinia Cousins married William John Justice (registered at Bicester, March quarter 1869; volume 3a page 757).
While I still haven't found Lavinia on the 1871 census yet (I haven't tried too hard though), she is on the 1881 census living at Tadmarton with her husband (a policeman) and 7 children aged from 1 to 11. The reference for the page is RG11 / 1525 / 66 / 7. Her birthplace is shown as Kings Sutton and her age (33) gives a birth year of abt 1848.
But where is the registration of her birth? I suspect it's the "Female Cousin" registered in the September quarter of 1848 at Brackley, vol 15 page 201, as I can't find a corresponding death registration for a female Cousin / Cousins / etc at that time. Lavinia must have been registered before she was named (the certificate might show the name added after registration in the final column?).
Hope this helps with that part of your mystery.
Steve