Basically, what was the diplomatic relationship between the parts of America under British rule and the newly formed republic? I have difficulties finding books that are not completely biased in one way or another. Especially from 1776 and up until the war in 1812.
There was a diplomatic relationship between the US and Britain after 1783 and the Treaty of Paris (where Britain recognized the independence of the US).
John Adams became the first US minister to the Court of St. James in 1785.
All diplomatic relations were between the US and Britain, not between the US and any colonies in the British North American Empire.
For a starting source, I would read Gordon Wood's Empire of Liberty: A History of the Early Republic, 1789-1815. It's in the Oxford History of the United States, which is generally written by top historians of the country. Wood is a relatively mainstream, conservative (in a scholarly sense, not necessarily politically) historian, but I think he's good for starting out on an issue, especially if you feel other books you've read are biased.
Read Bradford Perkins work, he is/was the leading authority on Anglo-American relations during the time period you specify