You seem pretty committed to London/Paris and that makes sense logistically, but I think you would have more fun just going to Barcelona or doing Madrid-Barcelona or Barcelona-Nice/Monaco
In London-
Do the tower of London tour, expensive but worth it.
DON'T do one of those garbage black cab or double decker tours
DO use the underground as it is super easy
DO go to picadilly circus and get pooped on by a pigeon
DO go to Harrod's
DO go to Westminster and the Royal grounds
DO go to the British Museum
If I can convince you to go to Barcelona I will give recommendos for that