I read a paper a while ago from Stephen Conway that deals with changing perceptions of the relationship between Britain and the Thirteen Colonies during and after the War of Independence. Bear in mind that while his perspective is useful in answering your question, it may not represent the full picture.

http://www.jstor.org/stable/3491638

Essentially, Conway argues that the British recognition of "Americans" as a distinct people (and not merely British colonials) had little to do with the Declaration of Independence, and that in general, the Declaration did not provoke a lot of response from the British public at the time. He surveys British newspapers in the weeks following the Declaration, and notes that while it was widely printed in Britain, it was rarely given prominence or commentary. The amount of response in the British press, whether in papers or in pamphlets was "unusually" small.

Conway attributes this general lack of interest to the fact that the actual event of the Declaration was not a surprise, and that in comparison to the treason of armed insurrection, the assertion of national independence seemed relatively minor. Government correspondence also indicates that the war, at this point, was still discussed in the language of rebellion, which is to say that the Declaration did not cause either the British public or the British government to regard colonists as anything but dissident British colonials with treasonous leaders.

On the other hand, those in favour of a conciliatory approach within Britain do seem to have been worried that the Declaration diminished the strength of their arguments, because it clearly outlined American claims to independence, not just to rights as British subjects. But ultimately, Conway comes back to his argument that while the assertions contained in the Declaration of Independence did attract the attention of the British public, the document itself did not - mostly because the ideas contained therein were already known to the British public. As such, it was not one of the motivating factors behind a British (or American) recognition of Americans as a separate people.