The chances are, given I am 38, that I will not be around in sixty years to see if I am right about this, but I do think Christopher Lee is wrong in his assertions! Britain's respect for the monarchy is not just for the Queen, but for the institution. It has respect that few politicians in this country receive or have received, save perhaps for Churchill. It has survived threats of invasion, social unrest, and returned after a brief period following civil war. Why? Because British Royals, well some of them, tend to be canny and know how to survive. They knew how to adapt with the times! Take Henry VII, Elizabeth I, Charles II, and more recently, King George VI They all faced threats to their position and were able to compromise where needed. Whether Prince Charles will remains to be seen, but Prince William is popular and may well do so!
But the monarchy may well be different in sixty years! Disestablishment of the Church of England and reform of the House of Lords will not sink it (although some seem to think there should be a single chamber, and damaging that would be!), but it may mean an end to the Coronation and instead a simple Inaugural Service, it may mean further changes to the Act of Succession (perhaps an election for the next monarch among the Royal Family like the College of Cardinals), but given it's survival so far and so long as the Monarch of the Day has a survival instinct, I think it is likely the monarchy will keep going!