I did some searching, but couldn't find an answer that's directly related: I'm curious as to what we know about the exact process by which the king found out about the Declaration of Independence. Would it have been given to him in written form by someone? Would someone have been tasked with telling him about it orally? If so, would he ever have read or heard the text in full?
I would also be curious about the Members of Parliament's method of receiving this information, along the same lines.
That's an interesting question. There doesn't seem to be too many records, buts let's see if we can put together a plausible chain of events.
John Dunlop printed approximately 200 copies on the night of july 4th.
Several of these copies made their way to England.
See https://en.wikipedia.org/wiki/Physical_history_of_the_United_States_Declaration_of_Independence
At least three are currently held in the UK national archives:
At the time, Vice Admiral Richard Howe sent two copies to the uk. The first with a letter dated July 28, 1776, and the second with a letter dated August 11, 1776
Packet ships (in this case, the Mercury) were among the fastest ships in the world at that time, so we can safely assume that the news carried by the Mercury was, if not the first to arrive in the uk, then at least the first authoritative indication.
Messages of this nature (by an Admiral) would have been dispatched directly to the English Navy Board, then recieved by the First Lord of the Admiralty. Since this was an actual Lord - at the time the Lord of Sandwich, one would think that the King would have been informed pretty damn quickly.
In addition, it's possible that George Germain, (1st Viscount Sackville) who was at the time Secretary of State for America (A British Ministerial Office) would have also seen a copy, sent by the British General William Howe. So again, Senior Ministers would have been aware of the contents within 3 to 4 weeks of the Printing. Lord North was Prime Minister at the time and would no doubt have asked for a meeting with the King to advise him of this. It's possible that there may be an entry somewhere in an offical document, that records this event.
The first known mention in the uk papers was the London Gazette on the 6th August 1776: https://www.thegazette.co.uk/London/issue/11690/page/1
The full declaration was printed in the English news by mid-month August 1776.
The first official acknowledgement by the king himself was when he addressed the British Parliament on Oct 31st, 1776. A copy of that speech is here: http://chapin.williams.edu/exhibits/His%20Majestys%20Speech.pdf
Hope this helps?