Can anyone recommend an online learning institution that supplies online A Levels with either Edexcel or Cambridge International?
All I have found online is companies that offer AQA and I'm unable to do that as during exam time I will not be in the UK and will need to do my exams internationally.
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Can't really give advice about CIE as it's not used by UK state schools (with a very few exceptions).
There are a lot of online resources for Maths so with those and the textbook there may not be a need to do a course. You need to make sure you're studying the right specification for where you are taking the exam - in England Maths has been reformed but I'm not sure Edexcel has changed its IAL specification. Same applies for Psychology.
English Literature might cause a problem. I don't know about CIE but UK specifications for English literature have non-examined assessment (aka coursework). It's very difficult to do these as a private candidate.