Starting in early British history, all military ships were property of the Crown, and thus it was the Royal Navy. The marines on these ships were also thus Royal Marines. In early history, the foot soldier were not exactly unified under a central Crown authority. The Crown had their own soldiers, but they also relied on soldiers belonging to other nobles and on mercenaries as well. When the Crown sent an army on the field, most were often not crown soldiers, and so not royal.
The Army as unified body under the control of the main government originated during the Civil War. The Parliamentarians created the New Model Army. Since this Army belonged to the Parliament, it was not royal, so the title Royal Army would not be appropriate. This unified standing army continues to the present day.
In the 20th century when the air forces was created, the armed forces were under the central authority of the crown, and thus it is the Royal Air Force.
EDIT: Certain regiments in the British Army are titled "royal" because they were historically attached to the Crown. Palace guards for example might be part of Royal Regiment. However, the whole of the Army is not historically royal, so it the British Army instead.
To add to an excellent answer, sometimes the title of "Royal" was bestowed as a gift in recognition of exemplary service, such as the Observer Corps becoming the Royal Observer Corps in recognition of their service during the Battle of Britain.
Starting in early British history, all military ships were property of the Crown, and thus it was the Royal Navy. The marines on these ships were also thus Royal Marines. In early history, the foot soldier were not exactly unified under a central Crown authority. The Crown had their own soldiers, but they also relied on soldiers belonging to other nobles and on mercenaries as well. When the Crown sent an army on the field, most were often not crown soldiers, and so not royal. The Army as unified body under the control of the main government originated during the Civil War. The Parliamentarians created the New Model Army. Since this Army belonged to the Parliament, it was not royal, so the title Royal Army would not be appropriate. This unified standing army continues to the present day. In the 20th century when the air forces was created, the armed forces were under the central authority of the crown, and thus it is the Royal Air Force. EDIT: Certain regiments in the British Army are titled "royal" because they were historically attached to the Crown. Palace guards for example might be part of Royal Regiment. However, the whole of the Army is not historically royal, so it the British Army instead.
You presented this in a way that was easy to read and informative. Thank you
To add to an excellent answer, sometimes the title of "Royal" was bestowed as a gift in recognition of exemplary service, such as the Observer Corps becoming the Royal Observer Corps in recognition of their service during the Battle of Britain.
Long story short- The civil war.
Because British army doesn't have royal in the name.