Packet boats, merchant shipping and naval vessels would all be used to carry dispatches and news to and fro between America and Europe; in the age of sail, you're talking about a timeline measured in weeks, not days — perhaps as many as six weeks in all.

General remarks on the transmission of news of the Declaration, from David Armitage's fascinating analysis of the Declaration and its global significance (emphasis mine):

Reports of American independence traveled immediately across the Atlantic Ocean and then deep into Continental Europe in the summer and autumn of 1776. Only two months after Congress had passed its resolution on July 2, word of inde- The Declaration of Independence in had reached as far east as Warsaw. The itinerary of the news illustrated the remarkable speed of communications in the late eighteenth century, as well as the richly developed network of newspapers and journals, and of spies and agents, that relied on the transmission of such information. Word had spread first to London, and from there to Scotland, Ireland, and Holland, before it was carried to the German lands, Scandinavia, and Southern and Eastern Europe, all in the space of barely eight weeks.

The text of the Declaration of Independence first appeared in London newspapers in the second week of August 1776. [pp. 69-70]

More specific detail on how news was transmitted to the British government:

The most efficient transmitters of the Declaration across the Atlantic were not the agents of Congress but British civilian and military officials in North America. During the autumn of 1776, these officers sent five copies of the Declaration back to Britain, where they later found their way into the British state papers. These copies now make up the largest collection of original printings of the document outside the United States. [p. 73]

Source: David Armitage, The Declaration of Independence: A Global History (2008).