How closely is the Queen's English linked to British royalty?

Yes, Queen's English, also known as Queen's Received English (QRE), also known as King's English and King's Received English in the days a male Monarch--Long live the Queen!--reigns, also known as Received Pronunciation (RP), is the English spoken by the Queen. Any assertion that all who abide on the Great Estate speak it is a ludicrous folly, for the Island of Great Britain herself has tens of distinct British accents and six major ones.

Queen's English, most commonly referred to as RP these days, is what many of the English call a "posh"* accent. Those who speak it naturally tend to be the extreme upper classes and those who did or do attend private boarding schools. Otherwise, those who don't come by it naturally but want to learn it, particularly actors and those who want to increase their social class/station, take elocution classes.

As for the technical end of it, i.e., grammar, semantics, syntax, etc., Oxbridge (Oxford and Cambridge) is the leading world authority. In regard to the grand English lexicon, the Oxford English Dictionary holds the keys and stewards on behalf of Her Majesty the Queen.

P.S. IN CASE I WASN'T CLEAR... Queen Elizabeth does speak Queen's English. Her accent is the accent. She's come off of some of it in her years, like no longer saying "house" like "hise" and "Paul" like "pool." If you pull up videos of her when she is young, though, she has that accent, the accent you will find affected in a lot in old movies, soaring. One does wonder, which came first, the chicken or the egg? Did she learn to speak like that or did everyone learn to speak like that because she spoke like that. By the way, the whole "Ma'am like ham"-ism is attributed to the Queen.

*The truly "posh" never call themselves "posh," neither do they call themselves "wealthy," for "wealthy" grossly understates it; they are "rich."