I'm a bit wobbly on Welsh geog - reply on Mr Google - who shows that Tregaron is a little place in Tredegar, Abergavenny

Bedwelty was in the registration district of Abergavenny, sub district Tredegar.
So - here's that family in 1851 - although I still can't spot them in 1861
1851
HO107 P. 2448 Fol 29 Pg. 724 Cast Place, Bedwelty
Dina Jones Head Wid 51 B. Cardigan, Tremain
Mary Jones Dau Un 21 B. Monmouth, Bedwelty
Elinor Jones Dau 12 B. ~ ~
Thomas Walters Son in Law M 25 Coal Haulier B. Glam. Merthyr
Margaret Walters Dau M 24 B. Cardigan Tremain
Rachel Walters Grand Dau 4 B. Monmouth, Bedwelty