I think there are a few issues here that should be separated for the purposes of clarity.
First — whether what we think of as the "Queen's English" is identical (or nearly so) to Shakespeare's English. There is a general consensus that it is not. People disagree about the details, but David Crystal is an example of a linguist that has spent a lot of time trying to figure out differences between modern pronunciation and Shakespeare's pronunciation.
Pronunciation and grammar have a lot of different parts. Although there is a widespread idea, referenced in the comments, that the grammar or pronunciation used by Shakespeare "survives" in the form of some present-day accent (often alleged to be Appalachian English), this isn't actually accurate. The most that can be said is that some features of Shakespeare's English that are not shared with most present-day accents of modern English might be shared with certain dialects of Appalachian English. But the presence of some shared features is in no way a guarantee of total identity between two accents.
The "Queen's English"
Second — what "The Queen's English" means, and where it comes from.
"The Queen's English" is a fairly abstract concept that is defined by how it is socially perceived. For simplicity's sake, it can be treated as an entity (a variety of English). But like many linguistic entities, it is not homogenous and its boundaries are not distinct.
The "Queen's English" has been distinguished from other accents by various features, some of which have changed over time. For example, it appears to have once been common to pronounce often the same, or nearly the same, as orphan, but it is now usual to maintain a distinction.
I don't know of any systematic feature of the "Queen's English", past or present, that was clearly invented abruptly out of nothing.
The closest thing to an "invented" feature of the "Queen's English" that I can think of is that I've seen some literature that alleges that the use of the sound /h/ in present-day standard English is substantially because of spelling pronunciation. Unfortunately, I forget where I read that: you can find a less extreme summary in Historical Phonology of English, by Donka Minkova (2014), which only goes so far as to say that "orthographic standardization, especially through printing after the end of the 1470s, was an important factor shaping the later fate of ME /h-/" (p. 107). In any case, while the use of /h/ has historically been a notable social marker in England, I hardly think it could be said to be a unique characteristic of the "Queen's English", and even if its use in southern English accents is largely based on spelling, the sound [h] was not invented as a completely new pronunciation of the letter H—rather, it seems to have been transmitted intact in at least some native vocabulary in other accents of English.
To return to the often/orphan example, I have never seen any evidence that the "orphan" pronunciation of often was established by someone who was consciously trying to create a new way of speaking. Instead, that pronunciation of often seems to have developed as part of a sound change that also affected other words with similar structure. Sound changes usually develop gradually and unconsciously. So I don't think it's really possible to identify the motivation behind the spread (and later decline) of this pronunciation.
It doesn't seem too implausible that some sound changes might have been influenced by pressure to have distinct ways of speaking for distinct social classes. But it seems very imprecise to use the word "invention" to describe that kind of process of unconscious differentiation between accents.
The idea that accents with low overt prestige (such as Cockney) originate from failed attempts to imitate accents with high overt prestige (such as the "Queen's English") is a generally discredited view in linguistics.