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 £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.
- Stansted - Berlin, on Ryanair (as mentioned by AntiSocialBehaviourOrder).
- Gatwick - Berlin, EasyJet, also very cheap.
- Heathrow - Berlin, British Airways (and other carriers). This will cost slightly more, but not necessarily a prohibitive price difference. There are many more flight times out of Heathrow, so this may allow you to make better plans to get the most out of your time.
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:
- Ciampino - Berlin on Ryanair, again very inexpensive.
- Fiumicino - Berlin a number of different carriers, more than Ryanair but not 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.