My Edney family were in North Curry, Somerset from about 1790.I do not know if there were any connections with your family before that, but I think it unlikely that there were connections afterwards.
There were quite a few Edneys around in Somerset and Cannington is only a few miles from North Curry so there could be a connection somewhere in the past.
I suggest you have a look at the 1851 Census for Cannington. I looked at an index and there are 4 references to Edney in the village, so one could well be your Thomas and his family. The reference is HO 107/1922 folios 95, 96, 118, 136 also folio 142 in Charlinch which is not far away.