I only started reading this a few days ago and am enjoying it very much, I do have two minor nits however,
1st.) The title should more correctly be ~ Captain Kirk and HMS Truculent, HMS standing for Her Majesty's Ship. The 'the' really doesn't belong. Using the words instead of the initials and it becomes quickly obvious. Captain Kirk and the Her Majesty's Ship Truculent. Although you could also call it Captain Kirk and the Truculent. But clearly HMS or the has to go.
2nd.) Contrary to what you may have heard, there was a great deal of sympathy for the American cause in England among the average man-on-the-street. Hanging the rebels would have had a drastic effect back in Britain with uprisings that would likely have crippled the government. A more realistic way to get the world you seem to be looking for would have been with a negotiated settlement that allowed the American colonies to send full delegates to Parliament as full members, not merely observers or guest speakers.
As I said the nits were pretty minor. On the whole this is an excellent re-working of TOS in a new setting. Please keep it up.