I wonder whether you have looked at any of the Oxfordshire Black Sheep booklets?
Some years ago I was involved in producing a name index to the whole series for the Oxfordshire Family History Society. This includes the following for William Eden (etc). The first number is the reference for the booklet concerned, the second on is page numbers(s).
Edden William 9 6 Calendars of Prisoners tried at Oxford Quarter Sessions and Assize Courts 1831-1835
Edden William 19 29 Banished ! Sentences of Transportation from Oxfordshire Courts 1787-1867 ; Surnames C to E
Eddens William 53 59 Childhood Secrets Volume 10 - children sent to Reformatories and Industrial Schools in Bristol, Cheshire, Essex, London, Surrey and Sussex
Eden William 3 20 Oxfordshire Constabulary Recruitment Register 1878-1904
Eden William 11 36 Calendars of Prisoners tried at Oxford Quarter Sessions and Assize Courts 1840-1842
Eden William 22 7 Banished ! Sentences of Transportation from Oxfordshire Courts 1787-1867 ; Surnames O to S
Eden William 39 39,67 Banbury Workhouse Records 1835-1843
Eden William 48 62 Wanted ! Military Deserters, Absconded Apprentices and Runaways 1802-1903
Edens William 53 59,60 Childhood Secrets Volume 10 - children sent to Reformatories and Industrial Schools in Bristol, Cheshire, Essex, London, Surrey and Sussex
Some of these look promising for explaining a disappearance - though not, I think the "childhood" ones.
Sadly, the booklets are no longer being printed. OFHS still have a few of them for sale. Otherwise you will probably need to find a library that holds them. The Oxfordshire History Centre should have them all.
Good luck!
Bill