This pertains to the army post Cromwellian era and before the Childer Reforms in the late nineteenth century. The British military organized their forces in the early modern era in four branches: Army, Artillery, Horse, and Navy. Of course, the Air Force came in later as its own branch in the twentieth century since it did not exist prior. The Artillery and Navy have royal distinctions since they are technically just one entity each. The Royal Navy has a long established history tied to the English kingdom and then the Act of Union in 1707 becoming the national navy. The army did not have such distinction. Now I have seen some claims that The Army had a royal status before Cromwell and during the English Civil War being that Charles II had the Royalist army backing him, but Royalist only pertained as a means of showing what side they fought for. I cannot speak for the Royal Navy and am cursory on their history, so I will stray a bit from their side since I do not want to post misinformation.

Historically, Britain, united or not, did not have standing armies for defense. Instead, Kings or other figureheads would raise regiments for service. At least until the seventeenth century. I would like to point out that the army still did not entirely function as one entity but had regiments. While the Navy and Artillery would be paid directly be the King, the army regiments had to be raised by Colonels - often times nobles and gentry. The mid seventeenth century saw the rise of some standing armies, but they remained to be separate regiments and not one entity. However, these separate regiments had the possibility to gain royal status by the means of clothing and title. For example, the Scots army had the First Foot, it became the Royal Scots once it became part of the English establishment in the later part of seventeenth century. By the eighteenth century, the redcoat had become the main uniform of the army while the Artillery and Navy had blue - denoting royal status. For the regiments that gained royal status in title, they added blue facings to their coats. The guards and certain regiments did so by the turn of the century. Regiments such as the 7th Royal Fusiliers, 8th Regiment of Foot, 21st Royal North Fusiliers, and 42nd Royal Highlanders had earned their royal status by a show of prowess in battle but had began under a different name and often of a colonel's name.

I hope another user can chime in, but the Royal Air Force and Navy had a charter designating their existence and establishment whereas the army did not when first created. The Childers Reform and the amalgamations in the later twentieth and early twenty first century changed a bit of the army establishment but still not entirely designate the whole army as Royal - keeping only certain regiments and corps as Royal.