Hello,

The answer to your question of when can appear for third year BA exam depends largely when have you done the re-registration for the third year, if you have given first year exam in June 2019 without any gap, that means you have enrolled in July session of 2018, and your re-registration date for the second and third years respectively comes in July session of the following years only in consecutive manner, as your course is annual based not semester wise, now it doesn't seem like you have done re-registration of your third year in last year of July session, if you had then you would have already filled the application form of third year for June TEE 2021 that is to commence from next month,

so to answer your question if you have done third year re-registration in January 2021 then you are eligible to appear for December TEE of this year, you will also be required to submit your assignments prior to it which you already know, if you haven't done re-registration till now, you can do it, last date is extended till 31st July, and you can accordingly appear in next year of June TEE, hope this clears everything, if not, you can ask me in the comment box below.