Paris to London ?

Definitely do it on your own. Pick a day and book your tickets ASAP. As more tickets are sold for a particular train the fare will keep increasing.

It's one hour earlier in London than it is in Paris. You will gain an hour going to London and lose one returning to Paris.

Buy one day, zone 1-2 paper travelcards with the National Rail logo on them when you arrive at St Pancras International. They cost £5.60 for off peak hours (after 09:30 on weekdays and anytime on weekends) and £7.20 if you want to take your first ride before 09:30 on weekdays. The advantage of travelcards sold by National Rail over those sold at Underground stations is that they qualify you for some good 2for1 discounts. See here. Print vouchers for the offers that interest you and take them with you.

If you don't have a problem walking a couple of miles during your London day, I would skip the public transportation option. Take a cab to your first destination, then walk. Check out the google London map for walking distances and wear comfortable shoes. If you do get lost, just head towards the Thames which is very central.

It's one mile from Westminster Abbey to the British Museum and you walk right by Parliament while the London Eye is directly across on the other bank of the Thames. Depending on which site you prefer, walk or take a cab first thing to either the Tower of London or Westminster Abbey, since both tend to have queues. After that, check a map and visit your other sites in order. There are two pedestrian bridges for crossing the Thames.

To save time, pick up picnic fixings at a Marks and Spencer Simply Food ( many outlets) and eat in Green Park --if it's rainy, find a pub instead.

With only one day, I would suggest actually entering the Abbey and the British Museum. Don't waste your time with changing of the guard---you don't see much unless you get there way ahead of time and then stand around getting tired.

If you do your trip on Thursday or Friday, the British Museum is open late those evenings. You could end your day there, then walk back to the nearby Eurostar station. There's a nice restaurant there or you could eat at the British Museum.

We have done all of these things on various trips to London.

St. Pancras International is the station for the Eurostar.

In one day I had time to:
Take a free (+tips) 2 hour walking tour of the City of London, walk to Leicester Square, have tea in the crypt at St. Martins in the Field, walk to Buckingham, and Westminster, take a trip on the Eye, dinner at Wagamama, a walk along the Thames back to the Tower Bridge finishing off the day with a Jack the Ripper Walk

While these may not all be on your list, it shows you can do a lot in one day, on your own with perhaps a little help from some walking tours.

Thank you all for your replies! You've inspired us to do it on our own. Booking the Eurostar tonight.