Essentially the British Army has it's roots in Cromwell's New Model Army fighting against the crown in the English Civil War. Furthermore over time it grew by absorbing regiments and other armies (eg the Scottish Army) each with their own histories and traditions. Some of these regiments in fact are royal such as the Royal Fusiliers.

In contrast the RN and RAF were formed by royal decree (Henry VIII and George V) respectively

Apologies for mobile post

This is answered here.

http://www.reddit.com/r/AskHistorians/comments/1c2qc4/why_isnt_the_british_army_referred_to_as_the/

The professional standing army was established during the interregnum and Parliament continues to have the prerogative to approve of the Monarch maintaining a standing army during peace time.

One thing that hasn't been covered in the thread thus far is the long-standing bias that the British political classes had against a standing army during the 17th and 18th centuries though until around the end of the Napoleonic Wars.

The rank and file had the image of being made up of the dregs of society, and it was a place to send 2nd and 3rd sons to become officers and keep them out of the way of the family. There was always a fear that having armed men around the place (not in barracks but billeted in inns very often before the American War of Independence) was a threat to civil society.

Thus both as an institution, and as individuals, the Army was seen as a necessary evil, but one that should be kept out of the way of decent folk and kept as small as possible, certainly not receiving endorsement by the Monarch as a whole.

related question: given these answers why are some specific units called royal [new word] e.g. Royal Fusiliers/RRF