The first state-to-state diplomatic proceedings happened on an informal basis during 1782-83, during the negotiations to end the War of Independence. Formal establishment of permanent diplomatic relations (and the recognition of the US as an independent state) didn't happen until after the signing of the Treaty of Paris in September 1783, which formally ended the war.

Ad hoc diplomatic negotiations between Britain and the United States began in the spring/summer of 1782. The British crown appointed David Hartley, as minister plenipotentiary (the formal title of an envoy or ambassador) and Richard Oswald as peace commissioner, to negotiate the cessation of hostilities. The American signatories to the treaty were John Jay (then Secretary of Foreign Affairs), John Adams (a roving US ambassador in Europe) and Benjamin Franklin (ambassador to France.)

A provisional treaty was agreed in November 1782; Congress declared a ceasefire in April 1783 and the final treaty was signed in September. Here's the full text of the treaty — in Article 1, Britain relinquishes its imperial claim over the colonies, and recognises the sovereignty of the United States:

His Brittanic Majesty acknowledges the said United States, viz., New Hampshire, Massachusetts Bay, Rhode Island and Providence Plantations, Connecticut, New York, New Jersey, Pennsylvania, Maryland, Virginia, North Carolina, South Carolina and Georgia, to be free sovereign and independent states, that he treats with them as such, and for himself, his heirs, and successors, relinquishes all claims to the government, propriety, and territorial rights of the same and every part thereof.

That's the starting point for formalised state-to-state relations. In May/June 1785, Adams arrived in London as the first US minister plenipotentiary to the Court of St. James. The British, for some reason, didn't appoint their first permanent ambassador to the United States until 1791, when George Hammond was given the job.