I entered "bremen to southampton 'nautical miles'" into a search engine and came up with this:

395 nautical miles

That, however, is by plane. The sailing distance would be somewhat greater.

Entering "steamship from bremen to southampton" brings up quite lot of information, including this very interesting site which might answer your questions:

http://www.gjenvick.com/SteamshipArticles/index.html#axzz2FW...