There are a lot of must sees in both London (an the UK) and in Paris. Depending on the time you have in London, I’d recommend looking at doing a day tour which gets you out of the city. I did a great tour which took us to Windsor Castle, Bath, Stonehenge and one other place I can’t remember. It as great to get out of the city and explore some of the other towns/cities out there. I did this all in one day on a bus tour and would highly recommend it if you have the spare day.
In Paris some places I would recommend would be getting to spend some more time in the Louvre, doing the Catacombs, seeing the Eiffel Tower at night, the Moulin Rouge and the Sacre-Coeur. There are heaps of things to see but these are some of the things that stand out to me.
If you really want to skydive then the helicopter ride above the valley was such an amazing place to do it. If that isn’t your bag of chips then Jungfrau is a good way to spend 3/4 of your day (if its a nice day up there). Also the Trummelbach Falls close to Lauterbrunnen are also a fun thing to check out. Especially if it’s a bit of a rainy day up on the mountain.
I hope this gets to you before your trip and sorry for the slow reply.