I am about to register for JOSAA Counselling 2020,as I have scored well in jee paper 2 conducted on September 2020. But due to other reasons,I have joined another college in my hometown. So when uploading my documents, especially TC,what should I do, because my original TC is with the college here.

I don't think you need a Transfer Certificate for admission into NITs or IITs. If it is needed you may visit your school to collect another. There won't be huge problem I guess. You need following documents for Document Verification. You need to upload these documents online.

FOR NIT+ SYSTEM

1. Class X Mark Sheet/Certificate, Aadhar Card (for Date of Birth)
2. Class XII Mark Sheet and Certificate (Qualifying Examination)
3. Certificate of category (SC/ST/OBC-NCL/GEN-EWS), if applicable
4. Original Certificate for Persons with Disabilities (PwD), if applicable
5. Crossed Bank Cheque copy / Image of Passbook with bank account details
6. Medical Certificate (in specified Format available on JoSAA-2020 format)
7. OCI/PIO card, if applicable
8. Passport (if applicable)
9. . Admit card of JEE (Main)

FOR IITs

1. Class X Mark Sheet/Certificate, Aadhar Card (for Date of Birth)
2. Class XII Mark Sheet and Certificate (Qualifying Examination)
3. Certificate of category (SC/ST/OBC-NCL/GEN-EWS), if applicable
4. Original Certificate for Persons with Disabilities (PwD), if applicable
5. Crossed Bank Cheque copy / Image of Passbook with bank account details
6. Medical Certificate (in specified Format available on JoSAA-2020 format)
7. OCI/PIO card, if applicable
8. Passport (if applicable)

Kindly go through JoSAA Business Rules before registering for counselling: https://josaa.nic.in/webinfo/File/GetFile?FileId=6&LangId=P