We are now in Reading, UK after having come back from Oslo and Tromso, Norway. I'll toss out lots of ideas based on what we accomplished knowing that you will need to pick and choose based on your time.
We flew from the US to London and spend 5 days traveling to Liverpool, Wales and back via rental car. Then we took a plane to Oslo for another 5 days in which we did a partial Norway in a nutshell trip up to the Flam area to do the train/fjord thing). After that we flew to Tromso (northern Norway) to see the northern lights. We will be spending another 4 days in this UK area to see some of London's tourist attractions. This is our second trip to England. Total trip = 21 days.
The Flam and the Vigeland Sculpture Garden are well worth doing and perhaps a ferry ride around the bay while there. But the Flam is a very long day because of the distance and not easy to get to from Oslo. It's better to take two days with an over-nighter in Flam. Be advised that the Flam with overnight package is not cheap.
You can easily drive over to Sweden from Oslo and travel down the E6 if you want the another country t-shirt. It's only a half day trip. You can take a whole day and do a ferry trip to Denmark. We did not because the ferry schedule did not match with our schedule.
You can fly to London or Germany for a couple of days. I love Germany. Europe airline hops are not expensive. But I would respectfully suggest researching the airline reviews unless you have a need to be packed in like a sardine. There is one in particular that I will never fly again. Fortunately, most flights are short.
If you choose London, then know that it's usually cheaper to use Gatwick than Heathrow. Use the train to get into downtown London and do a Hop-on-Hop off bus tour to see most of the attractions. I do not know anyone who hasn't been impressed with a tour of London's attractions.
A few things about Norway. First, it is expensive. As in 2 - 3 time more expensive than some other countries. Second, do not drive in Oslo during rush hour traffic. It is a miserable experience. Unfortunately, I mistakenly got that t-shirt. Lastly, you do not need to exchange currency for NOK. Everything can be purchased on plastic. Even a pack of gum.
Have a good trip.