Those are three rather challenging places to connect on a short trip. How many nights can they spend in Europe, not counting the night on the plane?
My guess is that it will be best to fly into London and out of Berlin, or the reverse, because there isn't a significant international airport convenient to Normandy.
I assume you already know that the arrival day may not be useful for serious sightseeing because of jetlag and possible sleep deprivation.
The Imperial War Museum is very large. I spent 8+ hours there over the course of at least 4 visits, and I was not interested in the military hardware. While in London they might also want to see the Churchill War Rooms (purchase ticket in advance to avoid bad lines), which I haven't see but am expecting to take over half a day.
It seems that the best path from London to Normandy is the Eurostar train to Paris (book far in advance for cheapest tickets), followed by regular trains to Bayeux or elsewhere in the area. That travel will take the better part of a day. Personally, I think travelers who are not WWII historians are best served by including a 1-day small-group tour of the invasion sites in their itineraries, then branching out on their own (with a rental car) to see museums, etc. Most visitors don't want to go to just one museum and a cemetery, and having a professional driver/guide allows for much more comprehensive coverage in a short period of time. The invasion sites are widely scattered.
Traveling from Normandy on to Berlin may turn out to be easiest by returning to Paris then flying. You can begin exploring the various options on Rome2Rio.com, but that's only good for high-level analysis. Use skyscanner.com for airline schedules and SNCF for rail schedules. Getting from Normandy to Berlin will consume most of another day.
I spent most of my Berlin time in art museums or at Cold War sites, but I know the city has multiple sights related to WWII, beginning with the Reichstag and the very large German History Museum. We have other posters much more familiar with WWII sites in Berlin and surroundings, including Dachau. [As Mark points out, Dachau is near Munich, not Berlin. Sachsenhausen would work.]
Berlin, like London, is a spread-out city. In both of those places it can take a surprising amount of time to travel from one stop to the next.
I think trying to see the three desired areas will lead to a very rushed trip for folks with a serious interest in this subject. Remember, the arrival day may prove worthless, and essentially two more days will be spent moving from one stop to the next. It's hard to imagine that any sort of justice can be done to WWIi sites if they have fewer than two full days in London, two full days in Normandy and three full days for Berlin/Dachau. That appears to call for a minimum of 10 days, including the arrival-in-Europe day and the two relocation days but not the departure-from-US or departure-from-Europe days.