When did Britain learn about the American Declaration of Independence?

The news reached London on the 10th of August. It was, of course, known by British officials in the colonies much earlier, but

It is astonishing how casually the Declaration was first reported to official London. On July 8 ex-Governor Tryon in New York wrote to Lord George Germain, the colonial secretary, and Admiral Shuldham wrote to the Admiralty Office, but neither of them made any reference to the momentous document. On the same day General Howe also wrote to Lord Germain, and buried in his letter is the brief sentence: 'I am informed that the Continental Congress have declared the United Colonies Free and Independent States.' That was all, as if he were reporting that Congress had recessed or had issued some new currency. These three letters reached London on August 10, making very good time crossing the Atlantic.

Source: H. Peckham, Independence: The View from Britain (Proceedings of the American Antiquarian Society, October 1975, Volume 85, Part 2)

As called2voyage and T.E.D. have pointed out in their comments, Mr. Peckham should not be astonished by this, and the author himself notes that

At least two pamphlets had appeared earlier in the summer predicting that the colonies were going to separate themselves from the crown.

Granted, the pamphlets were not quite the same thing but then the two sides were already at war (see also T.E.D.'s answer). On the declaration itself:

...by the middle of August a copy of the Declaration did reach London by some other means, as it was completely printed in The London Chronicle under date of August 16, although that issue of the paper appeared on the 17th or 18th. The British ministers and King George now all had a chance to read the forthright statement and react to it.

Source: H. Peckham

Also,

The St. James's Chronicle (London) for August 15-17 printed the Declaration in full, but with some curious editing to protect the king's name. In the second paragraph, where it reads 'The History of the present King of Great-Britain,' the wording has been changed to read: 'The present History of Great-Britain.' This omission of the king then makes it possible to change all the charges leveled against him from 'he' to 'it,' referring to Great Britain rather than the monarch.

On the reaction of the monarch and the government,

Official reaction did not yet manifest itself. Parliament was in its long summer recess, and most of His Majesty's ministers were out of London at their country residences. They did not feel compelled to issue statements, and as a matter of fact they looked upon the Declaration as a kind of dying gasp from a revolutionary force that was expiring. The military situation made them optimistic.... The rebellion couldn't last.

Source: H. Peckham

The British government reacted by hiring

John Lind, an English politician and pamphleteer, to write a rebuttal to the declaration. He wrote Answer to the Declaration of the American Congress, a reply that tried to pick apart the Declaration of Independence. Lind focused on the issue of slavery, saying that the colonists were actually angry that King George III had offered freedom to the slaves....

Following this, King George III officially declared the colonies to be in a state of rebellion. By August of 1776, the King ordered troops to the colonies.

Not until October 31st did the king speak before parliament:

In his address, the king spoke about the signing of the U.S. Declaration of Independence and the revolutionary leaders who signed it, saying, “for daring and desperate is the spirit of those leaders, whose object has always been dominion and power, that they have now openly renounced all allegiance to the crown, and all political connection with this country.” The king went on to inform Parliament of the successful British victory over General George Washington and the Continental Army at the Battle of Long Island on August 27, 1776, but warned them that, “notwithstanding the fair prospect, it was necessary to prepare for another campaign.”