To Everyone looking for Lydia Snowhouse
It is doubtful that you will ever find a christening record for Lydia Snowhouse under that name because the surname, "Snowhouse" never existed in England (or if it did, I can't find it anywhere I'd be very interested to know if anyone else has found it). I have gone through many books and websites on English surnames and I have just done a search of most of the major genealogy sites and Lydia is the only"Snowhouse"to be found anywhere. I became interested in this query because in many years of genealogical research, I have never seen the name, "Snowhouse".
I looked at the IGI entry and it was taken from Gibson's Oxfordshire Marriage Index (which does contain some errors), this was, I think, copied from Bishop's Transcripts, which were copied from the PRs. Lots of opportunity for errors. Of course, the vicar or parish clerk may have written the name incorrectly in the register. I think it is possible that Lydia told the vicar her name was "Lydia Snow House" and since Lydia was not "of the parish" the vicar was not familiar with her name and wrote it as one word instead of two. A mistake in the parish register or the Bishop's Transcripts, may be the most likely explanation as some of the information in online family trees seems to have been obtained from more detailed sources than Gibson's ( e.g. Lydia was "of Britwell Salome") .
Snow and House are both fairly common names in Oxfordshire and it is quite possible that Lydia's surname was House and perhaps her mother or grandmother was a Snow and so the middle name.It is also possible that she was illegitimate and her father was a Snow, as giving a child the father's surname as a middle name was common.
I would suggest that anyone trying to find Lydia's baptism record might start by looking at the original (or fiche/film) of that Pyrton marriage record (or have a reliable researcher do it) to make sure that the name is not the usual sort of mistranscription and really should be "Snodgrass"

. If it is "Snowhouse" or "Snow House", then search for baptism record of Lydia Snow House or Lydia House or even "Lydia Snow" (if illegitimate this is a possibility). This is probably a rather disheartening prospect, for it means searching again through all those records which you have already searched for Snowhouse, but I think it is the only way.
Of course this is only my opinion, but I thought I should offer it for consideration.
Good luck and I'd love to hear how you get on.
Carmela