Hey, this question (and many similarly like it) has been asked a bunch. [here is my answer to a similar one]
How long after July 4, 1776 did King George III receive the Declaration of Independence?
King and Parliament were likely to receive of the Declaration in late August or early September 1776 at the earliest. As far as I am aware, there isn’t a concise date between Revolutionary historians. That said, the official response did come on October 31st, 1776, however the initial response from the British government was Viscount Admiral Richard Howe and General William Howe about six weeks earlier, which you can see here. This document is rather short, and dismisses the ‘inadmissible claim of Independency” that the colonies declared.
The British Crown’s official response can be found here and like it was pointed out earlier, it was delivered on October 31st, 1776. The document asserts that King George III believed that the war was actually coming to a close and that everyone was almost ready to be friends again (which isn’t true), but that these lowly colonial leaders through a wrench in everything. Here, he says:
If their Treason be suffered to take Root, much Mischief must grow from it, to the Safety of My loyal Colonies, to the Commerce of My Kingdoms, and indeed to the present System of all Europe.
King George was hellbent on trying to assert that the colonies were ruining everything and that they threatened the might, grandeur, and happiness of the entire British Empire. This is a carefully crafted speech, one that was intended to rally Parliament into agreeing to fully-fund the war since they had been reluctant to take it seriously in the beginning and as a result, had lost some pretty key battles the previous year. It also served as a war-time update to inform the MPs that Washington and his Army had been thrashed pretty badly during the August campaigns, and why fully-investing themselves into the war was worthwhile. It was meant as a battle cry against these tyrannical leaders who were not doing the will of the people, but rather, they sought to selfishly distrust the Empire’s will.
Ultimately, the King and his advisors saw the Declaration of Independence as huge insult in addition to their treason. They wanted their military response to be seen in accordance of what they do to traitors (execution) and wanted to prove once and for all who held the power over these colonies.
So to specifically answer your question, no copy was sent from the colonists to England, however, several copies naturally made their way there and they were discussed in [newspapers by August 6th](https://www.thegazette.co.uk/London/issue/11690/page/1). Three copies are still kept in the UK’s national archives. For more information related to what the King’s response was, please feel free to click on the link above. If you have any follow ups, please post them here.
Edit: to clarify another commenter’s post. A common misconception about the Declaration of Independence is that it was created specifically to tell Great Britain that the American colonies had formally severed ties. While this is partly true, it’s main reason for existing was to declare it for the *states* to now be formally banded together. This was significant because it formally banded the American colonies together, in a way that no British colony ever had been, until this point. The leaders of the Independence movement also knew that there was no way Britain would recognize American Independence in 1776, so it wasn’t really worth sending an emissary there at that time. Even after the battle of Yorktown happened in October, 1781 which effectively ended the war, it took the U.K. nearly 2 years of intense negotiations to agree to recognize the American colonies as having been formally separated from Great Britain.
I’ve transcribed the Howe response below, in case anyone has difficulty with the image (or the stylized letter s):
By Richard Viscount Howe, of the Kingdom of Ireland, and William Howe, Esq; General of his Majesty’s Forces in America, the King’s Commissioners for restoring Peace to his Majesty’s Colonies and Plantations in North-America, etc. etc. etc.
Although the Congress, who the misguided Americans suffer to direct their Opposition to a re-establishment of the constitutional Government of these Provinces, have disavowed every Purpose of Reconciliation, not consonant with their extravagant and inadmissable Claim of Independency, the King’s Commissioners think fit to declare, that they are equally desirous to confer with his Majesty’s well-affected Suhjects, upon the Means of restoring the public Tranquility, and establishing a permanent Union with every Colony, as a Part of the British Empire.
The King, being most graciously disposed to direct a Revision of such of his royal Instructions, as may be construed to lay an improper Restraint upon the Freedom of Legislation, in any of his Colonies, and to concur in the Revisal of all Acts by which his Subjects there may think themselves aggrieved; it is recommended to the Inhabitants at large, to reflect seriously upon their present Condition and Expectations, and to judge for themselves, whether it be more consistent with their Honour and Happiness to offer up their Lives as a Sacrifice to the unjust and precarious Cause in which they are engaged, or to return to their Allegiance, accept the Blessings of Peace, and be secured in a free Enjoyment of their Liberty and Properties, upon the true Principles of the Constitution.
Given at New-York, the 19th of Sept. 1776.
Howe. W. Howe.
By Command of their Excellencies.