The Royal Navy are the senior service as they were the first to receive Royal recognition as an armed service of the Crown. The Royal Air Force were formed by Royal Charter in 1918.
The British Army is historically a collection of militias raised at the behest of the Monarch. Some regiments enjoy Royal patronage, others such as the Int Corps don't.
When the Monarch wished to raise an army he would do so by endowing a stipend to a local nobleman to raise a regiment. The nobleman would be the Colonel of the Regiment (hence that tradition of the senior officer commanding the regiment or division, or member of the Royal Family being the "Colonel" of the Regiment) and would determine its uniform, and promote its NCOs and select its officers from the sons of other notable families within the region. On raising a Regiment an oath of allegience would be sworn and a toast drunk. The army are considered "loyal" not "Royal" in that sense.
As we maintain the regimental system it is fitting that we are not the Royal Army, rather the British Army loyal to the Crown.