Interesting point. I never knew that.

The UK = as usual - is a mess of conflicting traditions.

There is no Royal Army, but all members of the armed forces, police, and certain prominent political appointments, swear an Oath of Allegiance to the monarch.

Note this is not to the country. And certainly not to the people. Not even to the Crown, which is a kind of abstract superclass of British monarchs in general.

The Royal Navy does not have to swear this oath. It exists by direct Royal Prerogative which means it's assumed to be loyal, pretty much by definition.

The individual regiments are very often royal titled though. "Royal Artillery", "King's Fusiliers" and so on.

Yes, it's complicated.

Indeed, my understanding is that the British Army was created by and legally answerable to Parliament, unlike the other armed forces. As you mentioned, a remain of mistrust after the monarch was restored to the throne after the civil war (though the navy did remain under the monarch's control). Also, there are royal units within the Army, but the whole of the army is not regimented in this way.