Europe in March with 12 year old

To me, that's way too much for 3 weeks. We visited for almost 3 weeks summer before last with my mother in law and three children (oldest was 11). We were in London for 8nights, Paris for 4 nights (our second trip there) and Belgium for 6 nights. For Belgium, we based in Ghent and visited Brugges and Iper with a guided tour of the battlefields. We spent our last day in Brussels before our flight. We visited Rome last summer for almost a week (part of a longer visit) and my children loved it. If your son loves history, you may want to consider London, Paris and Rome (taking a flight from Paris to Rome, possibly easyjet) If not Rome, I would choose one other city along with London and Paris. It will make for a really enjoyable time in each location.

The cities your son mentioned aren't all that close together, so you're right in wondering if you can design an itinerary that involves them all. I think I would take a step back and think about what kind of history your son particularly likes. For my boys, anything Roman is a hit (there are Roman ruins all over the place so this really isn't a limitation). They like castles. They are interested in battles. They didn't like churches at first, but I have dragged them to many churches and they now enjoy them as well (particularly when we tour towers, attics, tombs). World War II is always interesting. Once you have figured out what kind of history you want, then you can look for sites and destinations. These are not always big cities. Dover castle (Dover England) has plenty of WW II history, but also older history too. It's not in a big city, but you could do it as a day trip from London. We do a lot of history on our trips, but we try to stick to one country, or maybe two in the time frame you mention.

What a wonderful trip!

Your planned itinerary my be a little fast, but it's do-able.
You didn't say when you're going in March--but the later in the month the better the weather. And March weather can be little bit iffy, especially when you're wanting to go to places in the north. We try to go the last couple days of March to get in on the lower Winter airfares and do a swing to the south and work our way north.
I would suggest taking the Eurostar train from London to Paris. Then take a budget airline to Berlin. Trains can get you from Berlin to Prague via Dresden and it's worth a stop. Then fly from Prague to Warsaw on Czech Airlines.
I would suggest going a little farther south in March if it's your first trip to Europe--like Italy-Paris-London.

Do your 20 days include the flights to/from Europe? If so, you only have 18 days. You will use about 1/2 day for each change, so another 2 days gone. That gives you 16 days for 5 cities, 4 for London (because you'll be jetlagged on day 1) and 3 each for the rest. Don't underestimate the time it takes to get oriented to each new place - different languages, different transportation systems, different currencies. Do your 20 days include Easter weekend? Depending on where you are, sights may be closed.

I would narrow it down to 3 cities (and Warsaw wouldn't be one of them) with a day trip here and there.

What kind or era of history does your son like? (And what do the two adults like?) Use your more specific interests to narrow down what you want to see in each of those cities, and why - why did your son chose those cities? Especially assuming the 20 days includes your arrival and departure days, it's a little tight to visit all of those cities in one visit, especially if it's his first time to Europe. Also, do the three of you love cities;will you enjoy 20 days in mostly-large cities? Or would you prefer some time in the countryside?

Both of our kids (now 20 and 17 years old) like big cities; they also like art, history and castles. Each of their first trips to Europe, when each of them was 10 years old, we had a mixture of city and country, and was 2+ weeks long.

Our son loved, at that time, spies and stories about them. We began in London, including a LondonWalks tour called "Spies and Spymasters" (no longer offered). Then we took the train to Paris, picked up a rental car, and spent three nights in the Loire Valley visiting castles. We finished the trip in Paris.

Our daughter's first visit to Europe was to Italy. We began in Venice, then one night in Bologna. We then picked up a rental car and, after one night in one of the world's smallest countries, San Marino, we spent the next five nights in Le Marche, eating very well and visiting castles. We finished that trip with time in Rome.

We actually just returned from a short trip (during daughter's fall break) to Berlin and Prague - they're a good combination. We took the train in between, and spent one night on the way in Dresden. Much of Dresden is reconstructed, as you may know. But even knowing that, I found the city appealing. And we really like art (paintings), and their collection is quite good and had some pieces that I really wanted to see. So Dresden is not for everyone, but we liked it.

Warsaw, I think, is your outlier city. If you give one up, I would suggest that one.