It is quite likely he transferred to REs, they were primarily the main recruiters for Despatch riders when the war started, They would actively seek out recruits with motor cycle skills. It is also likely he would change regimental numbers on transfer.
Have you considered applying for his service records and medal information, he should have two sets having served in both wars, I know from this board that service records can take a long time to obtain but if you had them it would certainly make your research easier, his ww1 medals if any family member has them will indicate his serial number rank and regt.
James