Hi Rachel (for some reason your question doesn't appear to have come through properly, though when I click on 'edit' I can see it all.)
I don't have access to the Douglas Richardson books, but the genealogical chart in Farrer's Early Yorkshire Charters, vol 2, which is cited on the profile of John Bulmer seems to be correct (and is probably a source Richardson has cited in his books). The following pages, also cite primary sources for this family. KSB Keats-Rohan in her short articles about members of the 'de Bulmer' family in the volume Domesday Descendants, agrees with Farrer's chart.
The genealogical chart on p. 105 in Davison's Saxon Survivors, which has Allyn Bulmer as the father of John Bulmer, actually has quite a lot of errors, and the only mention of the early members of the family is this chart, there doesn't appear to be any other discussion that I can see. It apparently comes from a manuscript, but it looks very much like some of the genealogies found in various Visitations?