Hi Deb,
I think I can help with Francis and Sarah for 1861. They are presumably with Francis' parents Edward and Martha in Alford Lincs:
New Road;
Edward Elliott; Head; Mar; 60; Gipseys (" appears for Martha then all other professions are left blank); Notts, Willboughby
Martha Elliott; Wife; Mar; 69; Lincoln, Great Ponton
James Elliott; Son; mar; 30; Lincn, Grantham
Charlotte; Daug; mar; 28; Lincoln, Wyburton
Francis; Son; mar; 28; Leicester, Hellington Hill
Julia; Daug; un; 22; Lincoln, Timberland
Arnold; Son; un; 19; Lincoln, Bloxholme
Jermmina; Daug; 15; un; Lincoln, Barwell
Harriot; Daug; 12; un; Lincoln, Billingborough
Christopher; 10; Son; un; Lincoln, Potter Hanworth
Sarah; Daug; mar; 22; Notts, Nottingham
Thomas; grandson; 6; Lincoln, Cumbergate
Samuel; grandson; 4; Lincoln, Monthorpe
Lydia; grandaug; 2; Lincoln, Billinghay
William; grandson; 1 mo; Lincoln, Alford
John Smith; Boarder; 24; not known
No sign in 1871 though, also a shame it doesn't give more details of the boarder John Smith as may well have led to a connection between the Smith and Elliott families.