Give yourself 4 days in London as in reality it will only be 2 full days because of travel from the US and travel to Southhampton to catch your cruise.
Arrive Heathrow. Travel into London to your accommodation. Spend the remainder of the day getting acquainted with London via the Hop On and Off bus tour. Dinner in a pub or cafe near your accommodation. Fitful night of sleep.
Tower of London, be there at opening. 2-3 hours at the Tower. Walk from there to Leadenhall Market place, have lunch. From there walk to St. Paul's. From there tube to another part of the city. Are you shoppers? Museum goer's? Interested in Greenwich? Maybe this is the day you grab tickets at the 1/2 price ticket booth in Leicester Square for a theatre performance that evening. Or maybe you go online now to see what plays will be at Shakespeare's Globe in June and July, sign up to be notified when tickets go on sale and book seats for one of those performances that evening. After visiting St Paul's you could tube back to your accommodation to nap if you felt the need.
Start the next day with a Guided tour of Westminster Abbey then walk over to the Churchill War Museum. I think you are incorrect and tours of Buckingham Palace will be available in June and July. Double check this website in January 2016. https://www.royalcollection.org.uk/visit/the-state-rooms-buckingham-palace .
Next day Board your train to Southhampton.
As far as easy places to eat there are thousands of cafes, pubs, etc. Sandwich boards on sidewalks denote lunch and dinner specials etc. Pret a Manger's ( chain) has fresh soups, salads and sandwiches. Get some items to go and enjoy a picnic in any of London's lovely parks. St James, Hyde Park, etc. or find a bench along the Thames and enjoy watching all that happens on the River. Cafes in the Museums are nice and remember the majority of Museums are free.
London is a vibrant, bustling, historical world city. You need to decide what you truly wish to see and do there.
Lastly, visit a bookstore where you live and leaf through Rick's London guide. Really helpful to first timers.
DO NOT get the London Pass. Total waste of money.
Enjoy your quick visit.