I've not offered my two cents, but I'll add a couple -- given your limited time, do you have any specific interests that could align with suggestions? Like, periods of history, art, food, etc? I've spent a lot of time in both cities, as my spouse had extensive time in the UK as a consultant and we now own in the south of France (and our friend gives us access to a pied-a-terre near Notre Dame).
What part of Normandy are you interested in? Some people are interested in the D-Day beaches (usually with a trip to Bayeux), others want to see Mont St. Michel. There are other places like the charming Honfleur and the larger Rouen --- or to Giverny.
For highlights, I'd mention the Tower Museum, Buckingham Palace, the British Museum and the British Library but also theatre in the West End. I'd plan for a good (british) Indian meal and to take high tea. There are also some excellent other museums, like the V&A, natural history, Tate Modern, the Wallace Collection.
In Paris itself, besides the Louvre, Eiffel, Musee d'Orsey, Orangerie, Sacre Coeur, Notre Dame, Champs Elysees etc you might consider Versailles, the marces aux puces (Clingancourt), catacombs, Pere Lachaise, take a food tour and some of the specialized museums (Rodin, Picasso, Louis Vuitton, Atelier des Lumieres, Pompadou). Some of the better food choices are outside of the tourist-heavy 6th and 7th and into the Marais, Canal St. Martin, Bastille and even further into the 11eme.