You could tavel by train. Start in London then take the Eurostar to Paris. Then take an overnight train to Rome. Trains are very fast across Europe and easier than flying.
Some of my favourite places are:-
London- a boat ride up and down the Thames, London Eye, Tower of London, views of the Houses of Parliament, Buckingham Palace, St James Park, -easy to do in a couple of days.
Paris- Notre Dame, Eiffel Tower, boat ride on the Seine, Montmartre - maybe three or four days.
Rome- Coliseum, the Forum, Spanish Steps, Trevi Fountain and Ostia Antica (short, easy train ride). Also three or four days.