We lived in Europe (Germany) for four years and have traveled with pre-teens and teens.
When I first traveled my son was 5 and daughter 11. My son is now 49 and says he doesn't remember much of what he saw on trips prior to age 10. Museums are a bit difficult with young children. I remember when my son was almost 8 after half an hour in the Lourve he was ready to leave. Still, on that same trip he was in awe of The Sistine Chapel at St. Peter's and wanted to know all about the frescoes.
You can still do a lot with kids, but need plan more for toilet breaks. Also, children ages 6 and 8 might not have the stamina to walk 3 or 4 miles a day touring. Most importantly, when traveling with young kids, make sure you keep them close to you and don't have them carrying valuable items. My daughter lost her camera to a pickpocket in Rome at age 14.
Sometimes hotel rooms for five persons may be harder to come by, so you and your husband might have to split up if you get two rooms. I recommend B&Bs over hotels, especially if you visit smaller cities. In major cities like London and Paris you probably will have to go with a hotel. I have never rented an apartment, so I can't help you there.
I suggest a minimum of 5-6 days in both London and Paris, more if possible. Still, it is good to visit smaller cities and towns.
In England, I recommend a number of places that I have been fond of visiting. Here are some you can do with day trips from London (Windsor Castle, Stonehenge, Oxford, Cambridge and Blenheim Palace (Churchill's birth place). Also, Bath, Winchester, Warwick Castle (this is a great place to take kids), Canterbury and especially York (plan 2-3 nights there), Take the train for most.
In France, I recommend Verdun if you are interested in WWI history. Normandy is great, don't miss Bayeux and see the 900 year old Bayeux Tapestry. Strasbourg is on the border with Germany and opposite the Black Forest and near Switzerland. For the Black Forest consider Friberg and Triberg. Versailles Palace is a little west of Paris, but easy to get to by train.
In Switzerland (the most expensive country in Europe) you have magnificent scenic places. Consider visiting Interlakken and Lucerne. Geneva is nice. The lake district, that you mentioned is great and near Milan, Italy.
Bruges is great and Brussels is worth a day. Amsterdam you could spend a week, but mainly because of the Rijksmuseum and Van Gogh Museum as well as Anne Frank Museum. Anne Frank would not be advisable for young children. The art museums are great, but with three young kids, not sure how long they would handle them. We spent 3 hours in each.
From Amsterdam, do a Windmills and countryside tour.
I suggest spending two weeks in England, two weeks in France then at least a week in Switzerland. You could do another week in Belgium and Netherlands, if you have time.
Regarding beaches in Europe, you would be disappointed if you spend much time there. There are some good beaches in southern Italy, Greece and a few in Spain and southern France, but they are usually very crowded in the Summer, especially August. Also, they are not nearly as nice as what we have in Florida or California.
Disneyland, OK, I understand you have kids that would love that, but you can do that in the USA.
You mentioned the Amalfi Coast, but that is way south in Italy near Naples and it really has no beaches there. Italy is my favorite foreign country with all its ancient, medieval and renaissance history and art, but you seem probably won't have time to do Italy.
If cars are an interest, if you are near Stuttgart, the Mercedes and Porsche factories are there. You can tour them.
One thing my kids loved was going up the Eiffel Tower and doing a Seine River cruise in Paris.