You have little time to plan for this year and we have no idea of your interests.
The idea of four countries in two weeks is quite impractical. Would you cover India, Pakistan, Bangladesh and Sri Lanka in two weeks?
You need to study these places and select where to go, not try to do everything.
As to the children they would remember it as hellish, and probably tell you that often in all the years to come.
For two weeks pick two countries at the most with two cities in each that are close to each other. Personally, I would choose one country with three cities and spend the time exploring and not on trains and changing hotels.
You may impress friends at home with your whirlwind tour but you will not have a good time or impress yourselves or your children.
If you want such a tour then it would make sense to look for a bus tour. Even then I doubt if you would find one for four countries in just two weeks.
With kids we learned from experience that they do not do well traveling a lot to get from A to B to C... and changing locations and beds so often. Pick maybe 3 cities with not a lot of traveling between them, rent an apartment for a few days at each city so they can have a few meals at home with their favorite foods, get used to the same bed for a few days AND have time in the day to get to a park and play, especially if there are local kids also enjoying them selves.
You could look at central Europe where distances between France, Italy and Switzerland are not great but will give you a flavor of each country. For example if you fly into south of France like Nice or Marseilles you can easily visit parts of Provence, the French coast (beaches for the kids) and easy access into northern Italy and Switzerland. As you get away from the center, it gets harder to travel without wasting a lot of time and stressing/boring the kids.
If you want to add the UK, you really need to fly from Europe to save time, using one of the cheaper flights (EasyJet, Ryan,..) from a European city to probably London. Check the airline websites as they have interactive maps to show their routes. For example if you started in Rome you could then fly to London, or from Bologna fly to UK or Scotland, from Venice to London, and many others. Don't forget the Uk is not part of the visa free travel zone of Europe where you can just walk across the borders of Spain, Germany, Austria, France, Italy but could require a Visa and going through Immigration to arrive in the UK,
Four countries in two weeks is a whirlwind even if just two adults, with small children it's a crazy idea. And taking small kids on a bus tour is not the way to try to do it either. I agree with the others about reducing the number of places.
Italy and the south of France are very busy in August, on the coast as it is peak holiday time. Inland cities tend to be hot; this is why the locals are heading to the coast....which also means lots of the towns inland sort of "close down" in August with lots of shops and restaurants shut.. One part of Italy that is less hot, because it is mountainous, is Südtirol/Alto Adige, which is very beautiful.
We have spent time there in August and also in the Veneto, visiting Vicenza and verona and some smaller towns. We also went to Venice for the day.not particularly enjoyable as it was very hot and crowded. Our son was 9 at the time, I don't think we'd have done it if he had been much smaller (it's very difficult when kids are still in strollers because of all the bridges)Edited: 14 May 2018, 05:13
Lynda from the incredible Barcelona is right about the crowds in August, I missed the month in the posting. I guess it would probably be as bad along the Adriatic coast. Might be cooler up in the mountain areas of the north or in the Abruzzo Gran Sasso areas not far from Rome. I guess even Ostia near Rome on the coast would be crowded.
Stop....time to take a breath and reconsider as this is not a good plan, at all. You are wishing to visit 4 COUNTRIES - Italy, Switzerland, France and England - with small children and in 2 weeks during the worst month to travel - August. This is not feasible and will be a miserable experience for your family.
Does this look like what you had in mind? Fly into Rome 3 days/4 nights , then FLY to Zurich and take a train over to Lucerne for 2 days/3 nights, then FLY to Paris for 3 days/4 nights, then FLY to London for 2 days/3 nights and fly out. That's two weeks, four countries, four destinations. Do you actually want to be in 4 airports - 8 times - over two weeks?
August is PEAK high season in Italy and the weather will be scorching hot and all popular places extremely crowded. Rome, Paris and England are significant destinations and they take time to get to and time to visit. Switzerland takes time to get into the mountains and then back to a airport city. Do pay attention to what everyone is advising.
Or Munich in Germany (at least as simple as Malpensa....you can take a direct train from Munich to Bolzano/Bozen). You didn't include Germany, or Austria on your list but you could have a great two-week holiday in either of those too.
Four countries in two weeks may be best by a Tour as a taste.
We are doing France and Italy on our own July 29 to August 17th. Sorrento 7/30 - 8/3, Montepulciano 8/3 to 8/7, Rapallo 8/7 and 8/8. St Remy de Provence(rented an apartment for a week) 8/8 or 8/9 to 8/15 and Paris 8/15 to 8/17. Decided to rent a car at the train station Chiusi(near Montepulicano and drop off at Avignon TGV station. Cost a little more but better than other alternative.
So if you breakdown the time of each place. Three weeks is not that much for even two countries!