If you want to maximise the number of overnight journeys, you can take a Tallink ferry from Helsinki or Turku to Stockholm or a DFDS ferry from Tallinn to Stockholm, spend a day in Stockholm, then travel by overnight train to Lund or Malmö. You can't get from Lund or Malmö to London the same day but you can get to Brussels for a Eurostar the next morning or Hoek van Holland for the Stena Line ferry. The total would therefore be 3 nights travelling (or 2 nights travelling + 1 night in Brussels) plus 1 day travelling and 1 day in Stockholm.
If coming from Helsinki, it's cheaper and quicker to get the
train to Turku (17 €) and then the
ferry from there (from £62.73 when I tried 20th September). On 21st September there are tickets for
Stockholm-Lund/Malmö overnight for about £60 in a 6-berth couchette, £70 in a 3-berth sleeper or £150 in a single sleeper. These are non-flexible booked train only prices, the flexible tickets are about 20% more. The daytime trains are much cheaper. You can also buy a through ticket to Koebenhavn H which costs slightly more and includes the local train the following morning (change is in Lund). From Koebenhavn H you can book a ticket with
DB to Puttgarden in Germany. The Koebenhavn-Hamburg train goes on a ferry between Rødby and Puttgarden. It's normally much cheaper to book to Rødby on the Danish Rail website, then get off and get on the ferry, then back on a train, but I don't think there's time in this schedule. You can buy a Puttgarden-London ticket from DB with an overnight stop in Brussels (enter via: Liège and Lille to avoid the Düsseldorf-London bus). On 22nd September Koebenhavn-Puttgarden costs 29 € and Puttgarden-London costs 69 € (Sparpreise - book in advance, refundable for 17.50 € fee).