score:7
Flying is going to be much faster and much cheaper than taking the train. You can fly from London to Berlin for as little as Β£20 (!) round trip per person. The train is likely to cost at least five times as much. You may need to add fees for checked luggage, as well as travel to the airport, but still flying is almost certainly going to be cheaper. (If you go out and back to Berlin, you might consider asking your hotel if they will store a few bags for a few days, to travel lighter and save on fees).
There are a number of options for flying.
From London:
Both Gatwick and Heathrow are significantly closer to London than Stansted, so I would probably prefer one of the latter two options. There are even further options from London (City airport and Luton), but I think these three are the main ones.
From Rome, there are not quite as many, but still several viable options that aren't too expensive:
You could also fly from Paris on to Berlin, with a similar mix of options.
It would have been better to save back-and-forth travel by traveling London - Berlin - Rome or London - Paris - Berlin - Rome rather than going back to London. It may still be worth seeing whether you can change your tickets.
All that said, I would be wary of trying to do too much on one trip. I have made that mistake many times. London, Paris, and Rome is already a lot to see in 15 days. How much of the time do you want to spend travelling? And there are many interesting things to see near those cities, which you could see without adding so much travel.
Upvote:2
Paris-Brussels(-Amsterdam) by train is not particularly cheap but very fast and comfortable. If you also want to see Brussels or Antwerp, this seems like an option worth considering.
Beyond that, trains to Berlin tend to be slow(er) and do not seem very attractive as you are trying to cover a lot of ground in a short period of time. There are many things to see in Germany but Dortmund is not very high up the list so it does not make sense to take the train to stop there either. Instead, you could fly to Berlin from Brussels, Paris, or even Amsterdam and from there back to London or Rome.
Upvote:3
The budget airline Ryan Air fly from Stansted to Berlin for a reasonable price and in under two hours. Stansted is approximately an hour or so north of London by train so not hard to get to (travelling to Stansted from London is quite normal for flights so trains, buses and pre-booked taxis are plenty). Might be worth looking in to.
Upvote:4
High speed rail is a fun way to travel, and not unreasonably priced.
London - Berlin starts at β¬59.90 each way (about 9 hours): https://www.seat61.com/Germany.htm#London_to_Hannover_&_Berlin. The best way is changing in Brussels and Cologne.
I can find some Paris - Berlin trains on bahn.de from β¬49.90 (about 8 hours, one or two changes)
From Rome, you could take the night train to Munich and then the fast train to Berlin. Night trains are quite a time effective way of travelling, and can save on accommodation costs. Though probably not if you expect to have trouble sleeping on board.
The day train Berlin-Rome is about 14 hours (2 changes), so I wouldn't recommend it as convenient, but does take you past spectacular scenary over the Austrian Alps (https://www.youtube.com/watch?v=uucEm3op7uc) and allows you to change trains in Verona or Venice so has its own charm and might be up your street.