You did have to register a birth in the district it occurred in, but it seems that Annie was living in London at the time.
http://freepages.genealogy.rootsweb.ancestry.com/~framland/acts/1874Act.htm - number 6 probably applies here. She had her baby in Lechlade and then returned to Kensington. In order to register the birth she had to go to her local registrar, make a declaration and pay her fee. The Kensington registrar would have then forwarded the information to Faringdon who would record the birth in their books.
A good idea might be to look for the death and will of William Beaven who lived at Stock's House. He may have died in Hay district in 1912, aged "75" but ages at death are often off a bit.
About locating wills:
http://www.hmcourts-service.gov.uk/cms/1226.htmFor the 1881 census, you need to go into the 1881 Wales census on Ancestry, they have the English and Welsh censuses separate. When you're there, there will be boxes to put in the Piece, Folio, and page numbers at the bottom of the form. If you do that and don't put any names in, you should see the family come up.
