If you have a Fire TV Stick which is registered to your American Amazon account, you will also see apps related to the United States available inside the Fire TV Stick menu system. But, we all know that if you have the Fire TV Stick and the user registered in the UK, then you will see different apps connected to the UK available (such as BBC and more). But, if I want to get UK applications to my Fire TV Stick (sideload them) while using all the American applications as well, can that be done?
Based on my own testing the answer is that you can sideload for example the BBC iPlayer app from your Android device to the Fire TV Stick, but it will not work well, because this is not the app really made for the Fire TV Stick, but for Android devices. So, it will work and you can read more about sideloading Android apps to your Fire TV Stick here, but it will not work as good as the real Fire TV Stick applications do.
Another option could be to create a brand new Amazon.com user and register this with a UK address and so, and then use this user with your Fire TV Stick. But do not forget that you will need a UK IP address for this to work well with the UK apps. I have not tested this myself yet, but it sounds like something that will work, so I will test it and write more on this topic in the future!