If you file your 2020 return quick enough, the CRA might apply the tuition carry-forward to your 2021 return. If not, no worries. You can adjust your 2021 return to fix the amount.

Thanks for your reply.

While filing the 2020 return, do I say that I'm filing the tax credit for the first time? I've filed the 2021 return but it still hasn't been processed. 

Also, I thought I should just let you know, I used to live outside Canada in 2020 since classes were online due to Covid (I moved here in 2021). Should I file for 2020 (i.e. am I eligible)? Note that I'll be filing just for the tuition tax credit (no Canadian income, I don't think I'll be eligible for other credits/deductions)

If you haven't gotten your Notice of Assessment for your first return yet, then you still have not officially filed a return in Canada yet.

If you were a non resident in 2020, then you would have to mail or fax in your return for that year.