In London not only the Natural History Museum but also the V&A and the Science Museum are must-sees, and they’re all on the same road a minute or two’s walk from each other. Tip: avoid queues by entering the natural history museum via the earth galleries entrance. Also walk along the road on the way back instead of the subway (meaning underground walkway rather than tube) - it’s much quicker.
There are no good restaurants that I’ve ever heard of in that area. Go elsewhere to eat on that day.
There are lots of small museums near the British Museum, which I assume you have thought about visiting. The Petrie Museum is OK - basically like an offshoot of the Egyptian section of the BM - but the Grant Museum of Zoology is excellent for jars of pickled moles, etc. Nearby are also the Cartoon Museum and the Charles Dickens Museum, which you have to pay roughly a fiver for (the others are all free).
Near there is a decent and really old-fashioned chip shop - actually old-fashioned and run by a family, not pretend-old-fashioned. The Fryer’s Delight on Theobald’s Rd.
A 55 bus ride away is my favourite Korean restaurant, Tohbang.
St John’s in Smithfield is one of the best English (English food) restaurants, as long as you like meat.
Globe tickets probably will still be available. If you can stomach standing up for the whole performance, then groundling tickets cost £5 and are pretty much always available, particularly for matinees. Wear a rain mac with a hood - no umbrellas allowed.
From the Globe, a walk up the South Bank is unmissable for the views, the life, the river, etc.
Marks and Spencer, the supermarket chain, do AMAZING salads to take to the park with you. There are lots of parks that are worth visiting in their own right; it should still be warm enough for a quick stop.
In Paris, I’ve never actually been to the catacombs, but so many trustworthy people have recommended them that they seem to be a must-see.
Check the Eiffel Tower site before you go; it seems to be OK right now, but last time I planned to go it was right out, because the lifts weren’t working. You can also buy timed tickets in advance, so you could check out sunset times and plan to arrive at the top around then.