Oslo or Manchester

I would unequivocally choose Oslo , having been to both . Also , if you are still considering England , York would be a better choice there.

Have you been to Tallinn? You could take the ferry from Helsinki.

I have been to both cities and like them.

But, with everything the same, I'd choose Oslo.

I like both cities, too, but would tilt toward Oslo as long as you're OK with going to Norway and not really having time to see the gorgeous scenery.

But Laura's idea is even better, I think, if you haven't yet been to Tallinn. I think Tallinn has more zing for most travelers, and it's likely to be considerably cheaper than Oslo.

Stockholm is a nice city worth going back to and easy to combine with Finland. But if you don't want to return, both Oslo and Manchester will work and in the end very much depends on what you want to see and do. They will both offer what you are looking for and you will probably have fun whichever you choose.

But I'd also like to suggest Tallinn, amazing city well worth three nights. And plenty of options for day trips as well.

Frankly, I wouldn't go out of my way to visit Manchester.
Also, Oslo is not so close, why not Tallinn?

Thank you so much for these great replies! Oslo and Manchester because I was looking for a relatively easy and budget-friendly flights. I have been to the wonderful city of Tallin (some friends might be in the same boat as me and I will suggest they go to Tallin).
I will also look into York as suggested.
I was also looking at flying into London from Helsinki and going to Cardiff straightaway. However, the rigamarole of the National Express bus from LHR to London city center before going onward to Cardiff is a 5 hour trek (at least that's how a friend in Wales said he does it).
Thank you again, everyone, for the tips! :)

Since you mention you like museums. If you end up in Oslo, have a look at the Oslo pass. We really enjoyed having the 72 hour pass and thought it to be good value. It meant we could just pop in to lots of different museums and have a quick look at the things that interested us most. And there anre a lot of museums in Oslo! Also the pass included public transport which was easy to use - buses, trams and ferries - to get out of the city centre and also to visit some small islands. We very much enjoyed our 5 days in Oslo. Eating out was fairly expensive in Oslo but we stayed in an apartment and made some of our own meals (our preference in most locations as we tire of eating out).

If budget-friendly is important, you don't want to go to Norway. But have you considered Riga if you don't want to visit a country twice?