Many thanks to all of you for your help in this and in straightening me out. Sometimes it just helps to have another set of eyes looking at the same stuff and sometimes it just helps to have extra information!
So it looks like Frederick Callaway died of disease and had two army numbers 23589 and 1843640.
Cecil Callaway, although he was Frederick Callaway's brother, is not in the grave and survived into later life. In that respect he and Winifred is a bit of a red herring, although interesting in building the whole family tree.
It is Charles A Callaway who is in the grave and we have his Army/RFC/RAF records from FindMyPast (even if there is a typo and it gives his name as Galloway). Interestingly it says he was discharged to the RAF Reserve on 30 April 1920 which two years before he died.