Yes, you can write the NIOS exam a year after passing 12th standard. There is no restriction on appearing for NIOS based on the gap since your previous 12th board exams.  NIOS qualification is perfectly acceptable for JEE Mains. You just need to meet the eligibility criteria, which includes passing 10+2 with Physics, Chemistry, and Mathematics. JEE Mains considers the year of your first attempt at 12th board exams (NIOS or any other board) to determine eligibility. You can only appear for JEE Mains for three consecutive years after that first attempt year.

Similar to JEE Mains, BITSAT also accepts NIOS qualifications. BITSAT eligibility is based on the year you passed or appeared for your 12th board exams (current year and previous year). So, you can appear for BITSAT the year you take the NIOS exam and the following year as well.

https://www.careers360.com/exams/bitsat

I hope it helps!