1861 Census Bristol at Henry Row
Joseph Buswell (indexed as Baswell) 34 Engine Fitter born Chipping Norton Oxfordshire
Mary Buswell 33 born Westerleigh Glostershire
Frances 13 Waiscoat maker born Bristol
Thomas 9 scholar born Bristol
Charles Henry 7 scholar born Bristol
Emily 5 scholar born Bristol
1981 Census Bristol at Prospect Parade
Joseph Buswell 64 Engine Fitter born Oxfordshire Chipping Norton
Mary Ann Buswell wife 56 Born Bristol
It looks as though Joseph married twice
There are two marriages on www.FreeBMD
Joseph Buswell married in Bristol March Quarter 1846 reference 11. 189 to a Maria Beard
also Joseph Buswell married Mary Ann Saunders at Clifton 6a 377 December Quarter 1876
Looks like these may be yours.
I found Joseph on the 1861 Census by entering only Joseph and Chipping Norton and his age plus or minus a year or two
There was a Joseph Baswell born Chipping Norton so I new I'd found him.
It will be worth you ordering the marriage certificates.
You could look up the childrens births registrations on the freebmd site too.
There are very few Buswell names