I ventured to the library today. Tamworth doesn't hold St Paul, Fazeley anyway. Unless Pinetree comes back with an MI, then I will check the parish register for St Paul when I go to Lichfield Archives next week, although if GENUKI is correct I will only, possibly, find Thomas.
I did check Wilnecote Cemetery - both Old and New - and there is no sign of the Buxton family there, or Robina.
Maybe it's me, but I can't find any trace of Jane's death registration in 1912. Certainly not under Tamworth, or even Staffordshire.

Historical Directories do have an entry - 1870 Harrod & Co's Directory of Derbyshire, Leicestershire, Rutland and Staffordshire - image 848
Samuel Buxton, Esq
Thomas Buxton, Esq surgeon
Samuel Buxton's name also appears as a Bleacher.