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Here is a nice source for you. Although it is primarily about how Scandinavians got to America, I'm sure the info about the lines and routes that operated from Scandinavia to England can help you reverse the process..
Gothenburg was apparently Sweden's biggest port during this era. From there they would travel via steamship to Hull and/or Liverpool..
"In 1906 the Wilson Line formed a separate company with the North Eastern Railway Company to integrate some of their rail and steamship services. This new company, the Wilson and North Eastern Railway and Shipping Company, made even greater profits by shipping and then transporting by rail the thousands of emigrants they brought to the UK each year. The new joint company limited the numbers who travelled via any other shipping or railway company and ensured a degree of continuity in the journey from steamship to quayside not seen at any other UK port of entry. Although it was the Allan, Cunard, Dominion or White Star Lines who sold tickets throughout rural and urban Scandinavia to would-be migrants for travel to America, it was Wilson ships which brought almost all the migrants to the UK - thus generating huge profits for their owners. The Wilson Line was at the time the largest privately owned shipping line in the world and its size accounts for the dominant role it held over the migration of thousands of Scandinavian emigrants between 1843 and 1914."
Here is a few more:
Here is a more specific example of a passenger ship that apparently made regular journeys from Hull, England to Gothenburg, Sweden and back up until 1910.
"ARIOSTO / LUIS VIVES 1889
The ARIOSTO was a 2,376 gross ton ship, length 300.4ft x beam 38ft, one
funnel, two masts, a speed of 14.5 knots, and with accommodation for 53-1st,
24-2nd and 1,000-emigrant class passengers. Launched by Earle's Shipbuilding
& Engineering Co, Hull on 10th Dec.1889 for the Wilson Line of Hull (Thos
Wilson, Sons & Co).
"She made her maiden voyage from Hull to Gothenburg in
March 1890 and continued Scandinavian and Baltic voyages until June 1910
when she was sold to La Roda Hermanos, Valencia and renamed LUIS VIVES."
"When built she was the largest North Sea passenger ship on regular service
and was fitted with refrigerating machinery and electric light throughout."
In the link provided below you will find several ships that were built and utilized for Hull to Gothenburg service during the time period you are researching. It is believed that ships such as the Calypso, Ariosto (TIL 1910) and the Bayardo (1912) are just a few that ran this route
Other routes of passenger ships during this period involved Grimsby, England (east of Manchaster) to Amsterdam and then on to Gothenburg..