I figured out how to do this by using the Make Query Table tool, Copy As Python Snippet, the Python window and the Copy Features tool.
After running the Make Query Table tool to pull through just the fields I wanted to appear in the output I was able to Copy As Python Snippet this code from the Geoprocessing | Results window into the Python window of ArcMap.
arcpy.MakeQueryTable_management("'C:/avhome/data/WAregional/wa regional.mdb/schools'","QueryTable","USE_KEY_FIELDS","#","schools.OBJECTID #;schools.Shape #;schools.CODE #;schools.NAME #;schools.TYPE #;schools.Y11STUDENT #;schools.Y12STUDENT #;schools.COORDGEOCO #;schools.ID #","#")
and edit it to become:
arcpy.MakeQueryTable_management("'C:/avhome/data/WAregional/wa regional.mdb/schools'","QueryTable2","USE_KEY_FIELDS","#","schools.OBJECTID #;schools.Shape #;schools.Y12STUDENT #;schools.Y11STUDENT #;schools.NAME #","#")
Note that the new QueryTable2 retains the Shape field (so I can CopyFeatures it) and I have reordered the NAME,YR11STUDENT & YR12STUDENT fields. I also used the opportunity to drop out a few more fields.
The last step is to use the Copy Features tool on QueryTable2 which I did via its tool dialog to create a new feature class with the fields re-ordered permanently.