Best time of year to visit Paris & London

May is a good month for both Paris and London. It is not hot enough to restrict your activities (as it would be in southern Europe), but it should be warm (-ish) and (mostly) dry. The days will be long (longer than in Italy).
September, especially the second half, is autumn. You run the risk of cold, wet and damp, especially in London, and the days are a lot shorter than in May.

7 days for each city is good, it gives you time for the major sights plus day excursions from each city.

You're right to narrow it to May and September, and travel toward warmer weather. Chris is right about the choice between them. May wins for more daylight. Also, September is an expensive and busy month in Paris because of trade shows.

Not that there's anything wrong with April or October, we've enjoyed both, but you're increasing the chance of bad weather. Of course in the big cities weather is less of an issue than in the countryside.

A week in each city is a good introduction, with one or two day trips thrown in. You'll leave with a good sense of each place and, most likely, a list of sights to see on the next trip.

The best weather that we had in the UK was in September. The Gardens are still in bloom and you can even have some hot weather. Once you hit October...RAIN! ...which is OK in London which has fabulous Museums and Live Theatre. We were in the UK in April...Snow Flurries near the Borders. May: windy and rainy at Stonehenge...just dress for the weather and carry on. I've only been in France in the Summer which is wonderful with the long daylight hours. Fall might be wonderful in Paris at the Food Markets with the Mushrooms, etc.

Weather could be similar for both May and September, but of course is unpredictable regardless.
I would go for Spring (May) to get the longer days and the parks and gardens at their best.
Chelsea Flower Show runs 24-28 May 2016 with tickets on sale December 1st.
https://www.rhs.org.uk/shows-events/rhs-chelsea-flower-show/tickets
French Open Tennis at Roland Garros runs May 22 to June 5.

London & Paris are northern Europe, the weather is unpredictable. I've been cold and wet in July and sweltering in early June. This year I was in Paris for the first time in April and that's when I'd go back! All the trees were blooming as well as the gardens, beautiful everywhere, and not as crowded as later in the year.

This is the first time I have posted a query on this forum. I appreciate all of your wonderful and insightful responses. The helpfulness of Rick Steves travelers is remarkable! You have given my wife and I much to consider. Thank you and please keep the comments coming!

I would not worry about the trade show crowds in September in Paris. If you're taking the RS tour, you probably won't even notice any particular crowding. If you're traveling on your own, you just need to book your hotel well in advance. On the other hand, the third weekend of September is usually European Heritage Days in France and it can be terrific or terrible, depending on your point of view. On those 2 days, all the local and national government buildings are opened to the public and you can join the crowds to see marvelous places (palaces) other tourists never do. Nearly all the museums are free so they'll be really crowded, the worst time to visit them.

As Ben pointed out, one of the joys of Paris is just seeing it illuminated after dark. Sunset in mid-September is around 8 pm, in May it's closer to 9.30 pm.

We went to London & Paris two years ago. London first which had perfect weather (sunny, mid 70s), Paris second where it rained every day and was a bit cooler. I expected the opposite so you just never know. May is a great month to travel in though. Nicer weather, not too hot and a little bit less crowded than it would be in the summer months. Never been in September but I think that sounds really nice. Enjoy your trip whenever you decide.