Yes, you can appear for NET JRF and become an assistant professor anywhere in India — provided your Madrasa Board is government-recognized and accepted by UGC for higher education.
Since you’ve already been admitted to a government university for BA and MA, your eligibility is valid. You won’t face any problem during NET/JRF or recruitment, as long as your 10th and 12th marksheets are authentic and recognized. Just ensure your university is UGC-approved and keep all your original documents safe for verification.