Does "mid-June" mean that you'll be in London on June 13? If so, you should consider going to see the parade up and down the Mall to and from "Trooping the Colour," the official birthday celebration for the Queen. The ceremony itself takes place on Horse Guard's Parade at 11:00 a.m. but it requires a ticket, so you are out of luck for that. However, all the units (footguards, bands, cavalry, and artillery) march from their barracks, past Buckingham Palace then down the Mall to Horse Guards for the ceremony, then march back in review (complete with flyover and the Queen stepping out to the balcony to take the salute (she also rides a carriage to and from Horse Guards for the ceremony). Beats the heck out of changing of the guard.
When you go to the Tower of London, get there just before the opening so that you can buy tickets and beat the crowds. If you are there at the opening, you can proceed directly to the jewel house and see the crown jewels without having to wait in line. If you postpone this at all, you may have to wait in line for hours. After seeing the jewels, you can then come back to the entrance and pick up the free Yeoman Warder's tour.
Since you're not a UK resident, you can't tour the inside of the Houses of Parliament (foreigners only get to do that in September), but, you can go watch a debate in the House of Commons or the House of Lords, and you can do this at night (the House regularly sits in the evenings--check the calendar to find out precisely which evenings they will be sitting).
We took a day trip to Oxford and Blenheim. You can go to Oxford by train, but the cheap day return tickets aren't available until after 9:30, meaning that you won't get to Oxford until after 10:30. We took the "Oxford Tube" bus line (strange name, but it is a bus) and by leaving just after 7:00 a.m., we were able to get to Oxford before we could have even boarded a train for London. We were then able to take a local bus out to Woodstock (where Blenheim is located) and be there just before the Palace opened at 10:00 a.m. After doing Blenheim, we took the bus back into Oxford and got to see Christ Church College.