Epic 3 week Europe trip???

For a May/June trip of this year? Sorry to say you're not planning super early considering that is a busy time in Europe. Each time you switch a city you lose about a half day to travel (transport from hotel to station or airport, security process, actual travel time, etc.). Even with Eurostar you need to go through a security process that requires 90 minute ahead arrival time at the station in London.

I don't know much about the London or Paris Go City passes, but make sure to do the math to see if it pencils out for your family's interests. Convenience counts for some but if it doesn't cover what you want to see or sends you to sites you don't care about, it won't be worth the expense. Since you're conscious of budget, buy your Eurostar tickets 60-90 days ahead (as the system allows for your dates); the tickets only get more expensive closer to your travel dates. Be sure you use the official website and not a third party to buy tickets.

I recommend staying at a Disney hotel property at Disneyland Paris. You save a lot of time getting to/from the park and you also get in one hour early before the general admission. The park does dynamic pricing so check dates to see if there are more affordable options during your 7 day stretch. Download the park's app to assist with ride planning (wait times are very accurate). Pack snacks into the park to save $ on eating mediocre food.

Best of luck!

Updated it with 2027 thx.

"Pompeii and Mt. Vesuvius are a must for the family" have you considered flying from Paris to Naples? You can do Pompeii from Rome, but it makes for a very long day. If you get the extra time off take the fast train from Naples to Rome. For Rome you should plan on at least 4 days.

I've never met your lovely family but based on experience with our own offspring your children's ages (especially your daughter who if she's 17 now will be an adult by the time this trip happens) I feel it could be far more enriching now and in terms of inspiring their future travels if you were to subsitute other more locally-significant attractions in place of theme-parks/Disney, as given your location in the Western US I'd venture they've perhaps already experienced several including the OG.

1) skip the American stuff like Disney. Do the great European history.
2) 4 days each for London, Paris and Rome are not enough time. I have been to Rome twice for a total of 9 days and still not seen it all.
3) Pompeii is great, but you can't do it well on a day trip. Spend a night there and visit Sorrento and Capri.
4) Suggest you research exactly what you want to see in each city. TripAdvisor.com is a great source. Example, Rome, must see places are St. Peter's Bascilia, The Sistine Chapel, The Coliseum, Forum, The Pantheon, Trevi Fountain, Spanish Steps, Capitoline Museum.

You will need two days to see Venice.

Having a week in each city will be great.

I agree that you should do careful research before buying a sightseeing pass. I just took a quick look at the London Go City Pass. I see one of its covered attractions is the Hop On/Hop Off bus. Londoners actively discourage the use of HO/HO buses in London, because they spend so much time sitting in traffic. It's much less expensive to use public transportation.

London has a large number of world-class, free/donation-requested museums. One of those is the British Museum, which (though typically massively crowded) might be appealing to your children as well as to the adults in the party. That museum is huge, taking multiple full days if you want to see it all (which you obviously will not). If you go to the British Museum on a day covered by a sightseeing pass, it will chew up a lot of that day's sightseeing time, so you may not get so much benefit from the pass that day.

If you take a regular train to a sight outside London, you may be able to get 2-for-1 entry tickets there, which is a much better deal than any sightseeing pass. Hampton Court Palace is one place offering a 2-for-1 deal as of this year: https://www.daysoutguide.co.uk/hampton-court-palace. You can explore other possibilities on that website.

My general comment about big-city sightseeing passes is that to get the most benefit from them, you have to concentrate your high-cost sightseeing on the days when the pass is active, leaving free and less expensive sights for other days. What if there's an interesting inexpensive sight very near a high-cost sight? Are you going to skip over the former and make a second trip to that neighborhood on a different day? That wastes time, and time is the most valuable commodity on a trip to Europe.

I haven't been to Paris in decades, so I can't make useful comments about the Paris Go City Pass. Keep in mind that these are not passes put together by the local tourist office to encourage visitors to see a bunch of local attractions (you do find such passes in some smaller cities). They are money-making endeavors by private companies. That doesn't mean they would never save money for anyone, but the average visitor will lose money on such a pass. As Americans we are predisposed to expect buying in quantity will result in a discount. That doesn't necessarily apply to sightseeing passes.

In addition, if a sight requires a timed entry ticket, the pass will usually not get around that limitation, so you'll need to prebook an entry time online. At the least, you'll need to check the website of every major site where you plan to use the pass, to be sure there's no extra step required.

Suggest you spend 2 or 3 nights in Naples. Your visit to Pompeii will be enriched by a visit to Naples’ archaeology museum (MANN), which has ancient tile works, frescoes, statues and other artifacts such as glassware, locks and keys from Pompeii.

With two or preferably three nights in Naples, you’ll have a little time to dine or get pizza, take a passeggiata along the Spaccanapoli and not feel overly rushed trying to jam Vesuvius and Pompeii into a single day.

Four or five nights in Rome should work fine.

And, your kids are old enough that you can all skip the theme parks. So, perhaps take one of your three theme park days away from London or Paris and add to Rome and otherwise enjoy London and Paris without diminishing either. Totally skip Euro Disney or whatever studio theme park you had in mind in England. Save the theme parks for southern Cal, unless they’re adamant about going to European copies of what you have nearby. Let the kids actively participate in planning where to spend time in these iconic cities.

BTW, in advance of traveling to Pompeii, Robert Harris’ novel Pompeii is a fun read and should help put you in the mood.

One last thought. London, Paris & Rome all have those glorious BIG and famous sights with lots of crowds. Buckingham Palace, the Tower of London, the British Museum, the Louvre, the Eiffel Tower, the Sistine Chapel and the Coliseum and many more. But investigate the smaller sights, as well. Sites like Sir John Saone House, the Wallace Collection, the Old Bailey, Maison Balzac, St. Etienne-du-Mont, Basilica Santa Maria del Popolo and the market in Campo de Fiore. Apart from the RS guidebooks, look at DK Eyewitness and Lonely Planet for ideas further off the beaten track. My “kid” had to go to London for a few days for work and he spent a day doing his own graffiti tour of London that took in some Banksy stuff. Finally, leave some time and space for the unexpected discovery along the way.

I'd recommend working in a smaller location, as you have all big, bustling cities. Please consider Venice, it is a "theme park" in itself. Altho the San Marcos/Rialto Bridge areas are busy and crowded, it's not difficult to stay 15 minutes away from there, and experience the beautiful back canals and no crowds. Our favorite neighborhood is the Cannaregio area. Your family could take a group rowing lesson from Row Venice, very reasonably priced.
You have a lot of time to plan and it does get overwhelming, but you will have an incredible family adventure!