What external funding opportunities are there for American or British students undertaking a PhD in Sweden?
Higher education in Sweden is free at the PhD level, and free for EU citizens at any level. It is possible that an American who wants to take a Bachelor/Master level course as part of his PhD studies might need to pay a ~100$ application fee, but that should be it.  

PhD here is a full time job, so your living expenses will be well covered by the salary. It also means that you would start the application by finding an open position through university websites, or approaching a potential supervisor to create a new position. Additional scholarships are aimed mostly to cover the tuition fees for students who do need to pay, or to finance brief research collaborations.