The Declaration of Independence had little to do with Britain noticing the rebellion. In fact, the battles of Lexington and Concord happened in April of 1775, more than a year before the Declaration was written.

The National Archives has a decent history source on the Declaration of Independence located here, but as mentioned above, England was already well aware of the rebellion. A royal proclamation in August 1775 stated that the American colonies were in rebellion, and hostile action had already been underway for months.

Continuing with the same source (which is a pretty good one, given that they are the custodians of the Declaration, its drafts, and all the legal paperwork and congressional records related to it), the DoI was approved by the congress on July 4th 1776, and signing began August 2nd 1776, continuing for a while thereafter. By January 1777, they felt confident enough to publicly publish a second printing, with the names of all signatories attached (the first had just two names, to protect people from potential reprisals). All this would have made its way to England within not too much time, but didn't change much from the British perspective - it was pretty much a formal declaration by the colonies that they were trying to do something the king had already said they were doing a year prior.

I realize this doesn't answer the specific question in your subject line, but hopefully it satisfactorily responds to the part in your written explanation.