В каком смысле "напоминать"? Если в значении "показать, кто в доме хозяин", то однозначно нет: это было бы абсурдом. Великобритания уж точно не считает, что имеет какие-то права метрополии по отношению к США. Елизавета II принимала участие в торжествах по случаю 200-летия независимости США. Вот, для примера, её речь по случаю дарения мемориального колокола городу Филадельфии:
I speak to you as the direct descendant of King George III. He was the last Crowned Sovereign to rule in this country, and it is therefore with a particular personal interest that I view those events which took place 200 years ago.
It seems to me that Independence Day, the Fourth of July, should be celebrated as much in Britain as in America. Not in rejoicing at the separation of the American colonies from the British Crown but in sincere gratitude to the Founding Fathers of this great Republic for having taught Britain a very valuable lesson.
We lost the American colonies because we lacked that statesmanship "to know the right time, and the manner of yielding, what is impossible to keep."
But the lesson was learnt. In the next century and a half we kept more closely to the principles of Magna Carta which have been the common heritage of both our countries.
We learnt to respect the right of others to govern themselves in their own ways. This was the outcome of experience learned the hard way in 1776. Without that great act in the cause of liberty performed in Independence Hall two hundred years ago, we could never have transformed an Empire into a Commonwealth!
Ultimately peace brought a renewal of friendship which has continued and grown over the years and has played a vital part in world affairs. Together we have fought in two world wars in the defence of our common heritage of freedom. Together we have striven to keep the peace so dearly won. Together, as friends and allies, we can face the uncertainties of the future, and this is something for which we in Britain can also celebrate the Fourth of July.
This morning I saw the famous Liberty Bell. It came here over 200 years ago when Philadelphia, after London, was the largest English speaking City in the world. I was cast to commemorate the Pennsylvania Charter of Privileges, but is better known for its association with the Declaration of Independence.
Today, to mark the 200th anniversary of that declaration, it gives me the greatest pleasure, on behalf of the British people, to present a new bell to the people of the United States of America. It comes from the same foundry as the Liberty Bell, but written on the side of this Bicentennial Bell are the words "Let Freedom Ring"
http://www.nps.gov/inde/learn/historyculture/the-bicentennial-bell.htm