The good news is I went to Bedford Central Library today to look up your 1851 census index. The bad news is that it was printed in about 8 books containing surname indexes for each individual parish! I did however take a look at Clifton in case you missed something. From what you told me I am inclined to think that George and Ann died prior to 1851 which is why their children were being looked after by relatives.
97 New Farm (not sure about the N)
Samuel Green Head M 48 Farmer of 180 acres emp 9 labourers bn Bedford Standford
Lucy Green Wife M 50 Farmer's wife bn Clifton
Sophia Green Dau U 21 Dau bn Clifton
Samuel Green Son U 19 Son bn Clifton
Lucy Green Dau U 15 Dau bn Clifton
William Warner Green Son U Scholar bn Clifton
Arch Green Son U 10 Scholar bn Clifton
Jane Webb Servant U 20 House Servant bn Biggleswade
Benjamin Godfry Servant 20 Farm servant bn Clifton
Interestingly the Clifton index had Samuel's birthplace as STH meaning Southill. Standford is inbetween Southill & Clifton so may not have had a church of its own, in which case the inhabitants would have used either Clifton or Southill churches.
I checked the Clifton transcribed registers to see what I could find.
No BTs 1783-95
1784 Jn 14* (*registered) Mary d William & Elizabeth Green bn 3.5.1784
1786 (blank) John s William & Elizabeth Green bn 5.8.1756
1788 (blank) Edward s William & Elizabeth Green bn 5.11.1788
1794 Oct Sophia d William & Mary Green
1798 May 30 Arch s William & Mary Green
1800 Dec 33 Lucy d William & Mary Green bn 9.12.1800
1804 Mar 8 Martha Cole d William & Mary Green
I checked Southill transcribed registers thinking that Samuel may be George's brother:-
1806 Jan 2 Samuel s Thomas & Sarah Green bn 9.9.1802
1807 Jan 10 Mary d Thomas & Sarah Green
1807 Sep 13 Sarah d Joseph & Sarah Green
1809 Nov 30 Warner s Thomas & Sarah Green
1809 Dec 24 Sarah d Joseph & Sarah Green
1811 Nov 17 Ebenezer s Joseph & Sarah Green
1812 Dec 25 Eliza s James & Mary Green
Marriages - 1802 Aug 19 Thomas Green & Sarah Porter (Shefford) (lic)
1806 Oct 13 Joseph Green & Sarah Chamberlain.
Burials 1767-18012
1771 June 15 Ann wife John Green, tailor
1785 Oct 10 Sarah Green, shopkeeper, mortuary paid
1786 Dec 24 John Green, sen, tailor, mortuary paid
1804 Jan 9 John Green
Also on the 1851 Clifton. Census
25
George H Cooper Head U 60 Farmer of 183 acres emp 12 lab bn Clifton
William Arch Visitor U 52 Gentleman bn Clifton
Elizabeth Green Housekeeper U 61 House servant bn Reigate
Sarah Mitchenell Servant U 21 House servant bn Bermondsey
91
Sall Green head Widow 64 Annuitant bn Clifton
Marianne Green Niece U 29 Annuitant bn Midx Lambeth
Charles Tingey Servant 13 Servant bn Clifton
98
John Arch Head U 53 Farmer of 231 acres emp 12 men 4 boys bn Clifton
Mary Arch Sister U 58 bn Clifton
Mary Green Cousin U 54 bn Beds Old Warden
Rhoda Osbourn Servant U 16 bn Thurleigh
Amy E Smith Servant U 20 bn Dunton
Checked Old Warden PR baptism's 1739-1812 thinking Mary Green may be linked because of the Arch name connection:-
1797 Jan 15 Mary d Robert & Ann Green Farmer
1798 Apr 29 Elizabeth d Robert & Ann Green Farmer
1799 Oct 6 Lucy d Robert & Ann Green Farmer
1801 Apr 7 William d Robert & Ann Green Farmer
Confirmed George c 14.4.1805, bn 10.4.1805 s of John & Alice Green at Ampthill. Other children John c 8.3.1801, William c 25.12.1803, Ann c 21.5.1809, bn 11.2.1809 & Alice Hyde c 22.12.1811, bn 26.11.1811.
Confirmed Ann c 5.4.1801 d James & Esther Kitchener at Ampthill.
Checked for Warner Greens on the IGI when I returned & found the following, all at Meppershall (close to Clifton!).
William c 5.8.1795, George & Catherine both c 23.12.1802, Rebecca & Mary both c 15.10.1807, William c 3.8.1809 & Warner c 15.10.1807 all children of John & Sarah Green. Think this George might be worth thinking about or have you already eliminated him? The Arch link also might be worth checking out.
Can check the burials at Ampthill 1812 + in a few weeks time, in case the George there died before 1830 if you would like me to.
Keep searching
criggy