Cost Estimate for Trip to England and France

You would probably get more responses by going to the England / France forums separately.

In Britain, this chain offer good value accommodation - cheapest when booked well ahead:>https://www.travelodge.co.uk
A bit dearer is this lot:> http://www.premierinn.com/gb/en/home.html

I take it that you are using public transport. www.megabus.co.uk is the cheapest way of moving about Britain followed by www.nationalexpress.co.uk
However, the railways are faster and run to time (usually). You can get some cheap train tickets by booking ahead. Here is advice on rail travel:>https://community.ricksteves.com/travel-forum/england/how-to-save-money-on-railfares-not-pre-book
www.seat61.com

In London, the most expensive costs for you will be food, lodging and transportation and access to some temporary museum exhibits.
Transportation is complicated in London - so maybe someone else could explain it better than I can.
Look for hostels - some have private bathrooms - or budget hotels.
http://www.hostelworld.com/hostels/London

Purchase your Eurostar ticket 90 days in advance. You can often save a lot of money by travelling during off-peak hours:
http://www.eurostar.com/

In Paris, you can find clean budget hotels or hostels that suit your budget. The hotels will have private baths, but the hostels may not.
http://www.hostelworld.com/hostels/Paris

I would avoid AirBnB or any vacation rentals listed on the internet. You don't have enough time to investigate them properly. Low prices do not necessarily mean comfort in Paris.

Transportation costs in Paris - if you enjoy walking, that's free.

Otherwise, buy "carnets" - 10 tickets that anyone can use for Metro or Bus.
http://www.ratp.fr/en/ratp/c_21879/visiting-paris/

You can take the RER to Versailles - all the information for getting there from Paris as well as admission fees is here. Don't forget to punch your ticket in the machines on the platform:
http://en.chateauversailles.fr/

If visiting a few museums, you should compare the admission costs against the Museum Pass to see if it's advantageous. Make sure to check if what you want to see is included and will be open during your visit:
http://en.parismuseumpass.com/

Eating in cafes and restaurants can be costly. You might do better to pick up salads and vegetarien sandwiches in a bakery or supermarket for around 7 EU. Tap water is free.

These are great suggestions! And James, I took your advice and reposted this to the "England" forum - thanks!

You don't mention the dates for your trip this summer but here are some comments:

Big Ben and Westminster Abbey (anything else worth seeing nearby these? - can a peak of Buckingham Palace be easily seen around this area?)

Yes, you can always take a peek at the exterior of Buckingham Palace and it is not far from Westminster Abbey. In recent years the interior has only been open in August but it seems that the opening dates are more extensive for 2017, so you may be able to see inside on this trip.

Big Ben will be "silent" for several months this year for maintenance. As I understand it the tower will be sheathed in scaffolding for perhaps as long as three years. This is really unfortunate for millions of travellers like me who would like to get photos of the Houses of Parliament un-obscured by scaffolding. In front of the HoP is a statue of Winston Churchill (by Ivor Roberts-Jones). And in the HoP grounds (south of the main building) is a statue of "The Burghers of Calais" by Rodin. It's in a little park by the river and worth seeing even if you are not a Rodin fan. If you are a Rodin fan be sure to visit his museum in Paris when you are there. It has many of his bronze statues in the garden and if your legs are tiring it is a great place to rest them!

Tower of London and Tate (is that the best museum to see?)If you are talking about the Tate Modern I would say NO. I prefer the National Gallery for art. If you want a general museum I would recommend the British Museum. I think it's true that there is no entry charge for any of the London museums so, other than travel time/cost there is no risk attached to visiting any museum. If you do want to see the Tate Modern it is just across the Millenium Bridge from St. Pauls Cathedral which I recommend visiting.

what else do you recommend in London? The Imperial War Museum if you are interested in WWI and WWII history. Another must-see is Piccadilly Circus. But the two attractions at Piccadilly Circus are the statue of Eros and the mammoth advertising lights. I believe the lights will be "out" until the fall.

Windsor Castle then head to Cotswolds
I would take the train to Windsor and rent a car there (when your Castle touring is over) for your trip to the Cotswolds. I have always toured the Cotswolds by car but I understand that it is difficult, maybe impossible, to tour them by train. Be warned that some of the roads in the Cotswolds are very narrow. My favorite Cotswold villages are Bourton on the Water and Snowshill. I think that the RS England site photo is of Snowshill.