From an ex-New Yorker to a New Yorker, I can almost guarantee that you will NOT find London chaotic, crowded, or hot -- even on the tube, and even in our current "heat wave." I lived 50 yards from Oxford Street and found it quaint (and clean
).
The museums are almost all free. If there's anything you want to see, just go see it! If you go into the British Museum for five minutes to see the Rosetta Stone, you're not out any £. I have gone to the British Museum at least 10x (used to live down the street) and never had to wait in line for more than a few minutes. The line is usually for a bag inspection, so see if you can get by without a purse if you normally carry one.
I haven't been to the Churchill War Rooms, but the last few houseguests I've had have confirmed the crowds. If you want to go, book in advance for timed tickets, or prepare to queue for 90 minutes.
Re: food, I second the recommendation for Dishoom. Try to go in between lunch and dinner (either very late lunch or very early dinner) to avoid the queues, which can be 60-90 minutes on a good day. (Queuing is a national sport here!) Dishoom is Iranian café style and different from the usual Indian you get in the US, even in New York. Masala Zone is also nice, with more typical Indian food.
I would also recommend going to a Greek or Cypriot place, or going to anywhere that has halloumi, which is a delicious cheese that I have never seen in the US. (Maybe it's becoming more popular now?) Plus afternoon tea, of course: for casual, try Charlotte Street Hotel; for a little fancier, I like Amba Hotel.
Wagamama would not be my first choice
, but it's not bad. Do NOT - whatever you do - eat at Wahaca. The food is inedible. Mexican food in London is unspeakably bad and often contains mayonnaise in some form (on tacos, in guacamole....). The salsa usually tastes like ketchup, and the vegetables are overcooked. The food in London is generally much, much better than the reputation of the food in London -- but Mexican is the exception. It's even worse than you can imagine.
If you get a craving for pizza, check out Franco Manca or Pizza Pilgrims. The majority of the patrons will be speaking Italian.
Re: Paris, I agree that the Musee D'Orsay is more manageable than the Louvre, which is truly overwhelming in all respects -- size, crowds, etc. That said, the Louvre has some spectacular art, so make the trade-off. Unlike the museums in London, it's not free, though! I agree with Erin that Sainte Chappelle is incredibly beautiful.
I wanted to jump off the Eiffel Tower. Two hours in line, dealing with pushy tourists trying to cut the line (I can and did curse people out in four languages), and listening to stupid Americans moaning that their dinner reservation was lost only to find out that they made their dinner reservation at the "Eiffel Tower Las Vegas." (At this point, I switched from wanting to jump off to wanting to push this stupid lady off.) The views are great, but no how, no way, never again, especially in the heat. If you must go, get a reserved-time ticket.
Paris is the perfect place to find a sidewalk café, eat all of the bread and cheese you can, drink all of the wine you can, and try not to suffocate from the cigarette smoke that is everywhere around you.
If the smoke won't kill you, the BO on the metro definitely will. Worst smelling public transit outside of Philadelphia, if you ask me. Despite this, I do love Paris (but I love the rest of France a lot more).
Academic artists. If you do go and do like Impressionism, I would suggest starting in the Impressionist Gallery and then making sure to concentrate on those other artists that interest you. 
Newspaper headline (actual): "BRITAIN HOTTER THAN AFRICA." Reality: people walking around in sweaters (and Spaniards and Italians in puffy coats) in the evening.
