Tricky one, I cant see Lydia/Ann or William in 1851, nor a marriage
But 1861 gives a few clues. Seems they moved to Eynesbury after Edwin's birth, having started married life in Gamlingay. And William was not born a local man. I guess Lydia/Ann could have come from anywhere.
Is the surname for mothers maiden name clearly Breary on all the birth certs?
Not a name I am seeing on 1851!
Here is the 1861 to save anyone else who is searching having to look up the info
RG9 981 104 30
Eynesbury
William H Villiers 41 harnessmaker bn Staffs Burton on Trant
Lucy Franklin 24 servant housekeeper
Edwin H Villiers son 9 bn Gamlingay Cambs
Elizabeth 7 dtr bn St Neots
Georgina 4 dtr bn Eynesbury
George Franklin 1 son bn Eynesbury
I also note that Edwyn's birth reg is Oct/Nov/Dec 1851, so Lydia/Ann may have been pregnant with him at time of 1851 census (cant remember the date it was taken)