When travelling from London to northern Sweden by train, is it possible to have a day of free sightseeing anywhere other than Copenhagen?

The accepted answer is not working anymore as the city night line trains are taken out of the timetables.

So now your best bet will be to travel fast by day and/or travel by ferry at night.

Seat 61 suggests that you might use the Harwich to Hook of Holland ferry overnight, taking a day train to Hamburg, take a train the next day via Copenhagen to Stockholm and take an overnight train from there.

The German rail site offers several options for your travel Hook to Hamburg, the ferries arrival is scheduled for 8:00 AM, which would get you into Hamburg (with 3 or 4 changes) by 16:15 or 17:15, say it get you to your hotel by 5 or 6 PM, (17:00 to 18:00) giving you a good evening to explore and a good night of sleep. If you are lucky (and have 'every train' tickets which are more expensive) you might even make it to Hamburg just after 3 PM/15:10.

Hamburg to Sweden you can find on the same German site.

In Sweden you better use the Swedish site for the best results but the German site will still work and might be your choice.

There are several options. You can travel Hamburg to Stockholm on the day train, and go on to Östersund overnight. Or arrive in the afternoon and take a day train north the next day, again giving you a part of the afternoon and the evening to explore the city.

Some times of the year there are still overnight trains from Hamburg to Copenhagen, with good connections to Stockholm, but those trains are likely restricted to summer holiday season only as far as I can tell now.

The day train from Stockholm to Östersund takes about 5 hours, the overnight train, (leaving at 21:12 on the day I checked) takes about 9 hours.

Of course, timetables change, not just by year but also by day of the week and throughout the year.
Do check your intended travel before setting off. Most train sites include travel more than 3 months ahead, only when searching before the time table switches (early December and early summer) for trains after the switch you may not get results.

And as reminded by @Anonymous, there are a few ferries from Germany to Sweden, most of which can be reached by public transport on both ends, if maybe less convenient than rail travel through Denmark.
Do check while you do your planning but at this time I see ferries from Kiel to Gothenburg, Travemunde to Malmo, Rostock to Trelleborg and Sassnitz to Trelleborg, and I might miss one. I found them on this ferry search site.

The source of all knowledge on this sort of thing is The Man in Seat 61, in this case the Sweden page.

The bad news is that from their map, there doesn't look to be any suitable way to get from somewhere like Berlin over to Sweden.

That said, it does look possible to do Copenhagen -> Malmö -> Stockholm in an afternoon, rather than overnight. That would give you the evening and possibly some of the morning in Stockholm for sightseeing, rather than in Copenhagen.

Another option for the route is to take the ferry from Harwich to Esbjerg in Denmark, then on via Copenhagen. However, that doesn't look to get you much time anywhere else.

A final option is to do London to Hamburg in a day, then spend the night there (which would allow a bit of sightseeing). It's then a ~ 12 hour straight trip the next day to get you from Hamburg to Stockholm (via Copenhagan and Malmo). That might allow a bit of sightseeing in Hamburg, especially if you don't mind catching an overnight train for the Swedish leg.

Just a thought - since you have a route you know well, yet it still travels through several major cities, why not spend a night in Brussels or Cologne?

Brussels has great food to try, a fantastic building heritage and accommodation close to the station. You could even do a trip out to Bruges.

In Cologne, there's the famous cathedral right next to the station, and accommodation close by as well. The view from the top over Bonn and Cologne is fantastic!

If for some reason you don't want to stay a night in København, you can get a free day of sightseeing in Germany. I'm sure that you will get through Hamburg during your journey (as I know, this is the only railway to Denmark). This is a big city with many interesting places to go to.

You shouldn't be afraid about the time you will spend there - Hamburg is not very far from København, and getting to Malmö from there is easy - it only needs one change in København, and average time you need to get from Hamburg to Malmö is nearly 9 hours.

Also, you can search for a better train schedule, and get a day in Stockholm - it is a beautiful city.

I don't recommend going to Berlin as it is not on your route and you will spend more time on trains.

There are occasional night trains between Berlin and Malmö which use the ferry between Saßnitz and Trelleborg. These trains do not run everyday and not at all during winter, but could be nice for a change if you are travelling on a date when the train runs.