Pax Britannia, Mountbatten set that process in motion when he had the MOD created from the individual ministries, then it was stalled. Seconding RAF aircrew to the RN as the FAA ran down and more recently creation of Joint Commands was in vogue. Creation of a single staff college is another step. The problem is that everyone comes from a single service background with a better ethos than the other services.
You need real leadership to surmount single service prejudices. Ultimately a single initial training centre subsuming Sandhurst, Dartmouth and Sleaford Tech would be needed.
Then remember, the Canadians went the whole hog and now back again.