I've just unearthed a booklet I have, although it relates to Marriage Licenses and BT's, which might be helpful.
Marriage Licenses for Richmondshire are held at Leeds District Archives, so they might also have will and administrations!?
Alternatively the archdeanery of Richmond was in the Diocese of Chester.
"In Yorkshire, North Riding, the deaneries of Richmond and Catterick were in the eastern division of the consistory court of the archdeaconry of Richmond (diocese of Chester), the parish of Sockburn, the peculiar of Crayke until 1837, and the bishop of Durham's peculiar of Allertonshire until 1846 fell within the jurisdiction of the consistory court of Durham and the parishes of Brompton, Deighton, High Worsall, Kirby Sigston, Northallerton and Allertonshire and West Rounton formed the peculiar of Allerton and Allertonshire under the jurisdiction of the dean and chapter of Durham until 1846.
In Yorkshire West Riding the deanery of Lonsdale was in the western division of the archdeaconry of Richmond (diocese of Chester) the deanery of Boroughbridge was in the eastern division."
I'm assuming that as Wills and Administrations also came under the auspices of the clergy until the 1850s that the same areas will apply to them, as to marriage licences

but I could be very wrong.
BumbleB