Changing course at Oxford before starting?

This isn't impossible - normally there is space for an extra students in a subject (for instance one college may offer 6 places for Maths but only take 5 if one of them misses their offer). However given that fewer people are likely to miss offers this year given A-Level grades being predicted this may make things harder for you. A further complication is that you want to apply to a joint-honours course meaning that both the economics and history would have to agree to take you on that course. There are plenty of cases of people changing course during their studies but it's very difficult to comment on how likely your case is as it depends on each college. You should get in contact with your college academic office or admissions office - they will be able to provide an answer that TSR never will

Would the process be even trickier if I requested to switch to a course for which I haven't taken the specific entrance exam?

I'm an international applicant (Canada) and I received a spot in the Human Sciences course for this year. However, my interests have changed and I'm more interested in pursuing something Humanities related and Human Sciences at Oxford is more science-leaning and not as interdisciplinary as I thought it was. I really want to go to Oxford but I don't think I would be happy in my current course. Do I have a shot at switching into a course like History and Economics? I did well on the TSA if that helps and my interviews were good. I did AP exams in History and Maths so I would be a good fit. How could I approach them in order to change my course? Is there a process?

Contact them sooner rather than later (the academic office at your college would be your best bet). I would not get your hopes up - changing course is possible but it is uncommon and not just an administrative exercise, and I think that @Nick125 raises a very important point - the cases that I know of people switching were from more competitive courses to less. By all means contact them, they won't withdraw your offer for that, but expect that the answer will be no and start thinking about what you'll do if that's the case.

I contacted them and told them I would be interested in changing my course. History tends to have a higher acceptance rate to Human Sciences and I would happily ask for History. I very much enjoy History but I am worried because I have only studied American History as an AP (like an A Level or HL) and only have general knowledge about European and World History from regular classes.

I would imagine that, if anything, it would be the economics part it would be harder to switch to, as it is such a popular subject and tricky to get in for.

What did they say to you? Is it possible to switch to history? Will it depend on how many meet their history offers?