I can't find any overnight trains from London to Berlin. Only ones that have a bunch of connections, which would mean I would have to travel during the day. Is this actually the case?
Unless I'm proven wrong by another answer, then yes, for the most part. There are some options though.
If you look at the UK version of the Eurostar website, you can book tickets to a variety of places in Europe, but in Germany - only Cologne and Aachen. Even these only take 5ish hours, and are during the day, as far as I can tell.
They offer the following suggestions
There are several good options for travel between the UK and Berlin or Hanover.
So at least there with option 1, you'll get to travel much of it at night.
Loco2 - loco2.com, which is my company, has worked hard to make booking this route as easy as possible. We combine Eurostar trains with the City Night Line sleeper train from Paris so that you can easily book all in one go. We also offer other options via Brussels if you want to look at day trains as well.
All the prices are the same as booking directly with Eurostar, SNCF or Deutsche Bahn. The combinations offered on Loco2 can't be found anywhere else because of the way that we've built our software (you can book each portion separately elsewhere but you'd need to manually combine the different searches as suggested by Seat 61).
Sorry to sound like a bit of an advert! Hope that's helpful, and I'm happy to answer any more questions about Loco2 or any other aspect of how best to book London to Berlin trains.
There are currently no sleeper services operating through the Channel Tunnel, so no sleeper services between the UK and Europe. They originally planned to run some, and had the Nightstar trains built, but for various reasons detailed on the wikipedia article they never ran. I believe you can try the rolling stock if you go to Canada!
So, short answer - there's no sleeper rail service between Berlin and London
The every brilliant Seat 61 page on the journey lists your main options. If the sleeping bit is important, opt for a train to the Netherlands then change for Hook of Holland, then take the overnight ferry to Harwich. Those have nice comfy beds, small but decent showers, wifi, restaurants etc. You'll sleep much better on that than you would've done on most sleeper trains!
Otherwise, if you want to spend most of your time sleeping on a train, then you'd probably need to take the Berlin <-> Paris sleeper, and change to a Eurostar. Seat 61 has the details on that option. Oh, and if you're googling for other options, you may see references to a sleeper Berlin <-> Brussels with a Eurostar connection. That used to offer excellent timings, and was good to take, but most annoyingly it hasn't run for a number of years now and is apparently unlikely to restart (disputes between DB and SNCB over track charges I believe).