I'm assuming that the Daniel and Ann Oakes living in Garston in the 1871 census are your couple, luckily must have just married before the census was taken.
I suppose the death in Kidderminster means that maybe she went home to family to be cared for when she was sick.
In three years there's every chance that there was a child, could be even that she died in childbirth.
The main suggestions I could give you would probably involve spending money, depending on where you live and you might not think it worth while for something that is unknown.
If you check the birth indexes and list any Oakes born in Liverpool or the Kidderminster area (particularly towards the 1874 mark) and try to identify whether they might have belonged to Daniel and Ann by perhaps checking baptism records. So of course if you can't get to the Liverpool Record Office it would mean ordering Parish Registers through the LDS.
Living alone in 1861, if I have the right Ann, would seem to indicate that she didn't have any parents at least, although there might be some siblings about who might possibly have taken in a child, but it seems unlikely to me.
Looking at the 1881 census gives the one clue that I would follow up on - your ggrandfather's elder brother Thomas Walter. If he is aged 7 in 1881 it's questionable which of the ladies would be his mother without the birth certificate. However, I don't see a birth for him on FreeBMD in Lancashire and the only one in Worcestershire is in Dudley for a Thomas Walter is dated 1876 which seems a bit late to me.
I think you should check the indexes themselves rather than relying on FreeBMD being accurate, and make certain that there are no Thomas Walter's in either the right area of Worcestershire, or .... I believe West Derby is the right area to cover Garston, and then check for a baptism. As Daniel is still living in Garston in 1881 the chances are that if you have found William's baptism then the other children might well have been "done" in the same church, and see what the mother's name has been given.
Good luck