You have a few options:
Option 1: Rail from London via the Harwich Esbjerg Ferry.
Option 2: London to Cologne plus night train to Copenhagen.
Option 3: Daytime train all the way.
Option 1: The Ferry
The DFDS Seaways “Dana Sirena” departs from Harwich at 17:45 but you need to get there from 16:00 to make sure you check in early as this closes at 17:15. This means taking ideally the 14:18 train from London Liverpool Street which gets you in to Harwich at 15:49. The next train is 15:18 which means you would only have about 20 minutes to get off the train to check in, which is not worth the extra stress. The off-peak return costs £28 but we recommend you book this train as on Fridays it can shoot up to £56, especially on a bank holiday. We find thetrainline.co.uk the most useful method. You might find that driving is more convenient for you, in which case there is secure parking right in front of the terminal at £7.50 per 24 hours, not per day, as is the case at BAA Airports, for example. There is always plenty of space there, and you need only turn up, take a ticket and pay on return.
The ferry journey takes about 18 hours and arrives at Esbjerg at 13:00 the next day. At booking, you must take a cabin, you cannot share with strangers, so obviously it’s best to take a travel partner. You then have a choice of inside and outside facing cabins, with upgrades to the very nice Commodore Class well worth the extra money. You get access to a private bar, sky TV, your own lounge at the top level of the ship, and breakfast included.
The train station in Esbjerg is another 5 minutes walk from the ferry terminal (and a lot more tranquil than Liverpool Street!). From here you can travel to onward destinations in Denmark. You check train times at www.dsb.dk.
Option 2: The Night Train
This type of journey involves booking a Eurostar train from London to Brussels, connecting to a Thalys train which would then take you to Cologne. From Cologne you can get a City Nightline sleeper train to Odense or Copenhagen.
The Eurostar can be booked direct with the Thalys included, but you can also get the same tickets on the Rail Europe Website where can also book the City Nightline.
As for timings the City Nightline leaves Cologne at 22:28 which gets you in at Copenhagen at 10:03 the next morning. You really need to be arrived at Cologne by 21:00 for safety, so we suggest a 12:57 Eurostar from St. Pancras which, by the time you change at Brussels Midi will get you to Cologne at 19:45. You will have some waiting time, but if you want to avoid that try the Rail Europe website where you can book the German ICE trains from Brussels instead.
Option 3: Daytime Trains Only
Can’t be done in one day, so if you want to see all the views en route to Denmark, the only real option is to stay a night at a Cologne hotel.