Arnav, this is the excerpt from the Information bulletin that you are talking about.

The Higher/Senior Secondary Examination or the Indian School Certificate Examination which is equivalent to 10+2 Higher/Senior Secondary Examination after a period of 12 years study, the last two years of such study comprising of Physics, Chemistry, Biology/Bio-technology (which shall include practical tests in these subjects) and Mathematics or any other elective subject with English at a level not less than the core course for English as prescribed by the National Council of Educational Research and Training after introduction of the 10+2+3 educational structure as recommended by the National Committee on Education. Candidates who have passed 10+2 from Open School or as private candidates shall not be eligible to appear for ‘National Eligibility Cum Entrance Test’. Furthermore, study of Biology/Biotechnology as an Additional Subject at 10+2 level also shall not be permissible.* *The candidates who have passed 10+2 level with Biology as an additional subject will also be eligible for MBBS Entrance Examination (as per Hon’ble High Court Order No. 2341/-W/DHC/WRIT/D-1/2019 dated 24/09/2019 in the Writ Petition (C) No. 6773/2019. If the institute has to consider an examination of an Indian University or of a foreign/university to be equivalent to the 12th Class under 10+2 scheme/intermediate science examination, the candidates shall have to produce a certificate from the concerned Indian University/Association of Indian Universities to the effect that the examination passed by him/her is considered to be equivalent to the 12th class under 10+2 scheme/intermediate science examination. *(NOTE: The proviso in italics has been subject matter of challenge before the Hon’ble High Court of Delhi, Hon’ble High Court of Allahabad, Lucknow Bench and Hon’ble High Court of Madhya Pradesh at Jabalpur. The provisions of the regulations disqualifying recognised Open School Board candidates and the candidates who have studied Biology/Biotechnology as an additional Subject has been struck down. The Medical Council of India has preferred Special Leave Petitions before the Hon’ble Supreme Court and Appeals in the Hon’ble High Courts. Therefore, the candidature of candidates in the NEET (UG)-2020 who have passed the qualifying examinations i.e. 10+2 from National Institute of Open Schooling or State Open Schools or as private candidates from recognised State Boards; or with Biology/Biotechnology as additional subject shall be allowed but subject to the outcome of Special Leave Petitions/Appeals filed by the Medical Council of India)

Now we will quote few things in Bold and Italic for your reference:

The proviso in italics has been subject matter of challenge before the Hon’ble High Court of Delhi, Hon’ble High Court of Allahabad, Lucknow Bench and Hon’ble High Court of Madhya Pradesh at Jabalpur. The provisions of the regulations disqualifying recognised Open School Board candidates and the candidates who have studied Biology/Biotechnology as an additional Subject has been struck down. The Medical Council of India has preferred Special Leave Petitions before the Hon’ble Supreme Court and Appeals in the Hon’ble High Courts. Therefore, the candidature of candidates in the NEET (UG)-2020 who have passed the qualifying examinations i.e. 10+2 from National Institute of Open Schooling or State Open Schools or as private candidates from recognised State Boards; or with Biology/Biotechnology as additional subject shall be allowed but subject to the outcome of Special Leave Petitions/Appeals filed by the Medical Council of India)

This means that the NIOS and Private Candidates will be eligible to appear for NEET but their admission will be anytime subjective to the Pending case in the court. So until the Supreme Court passes its judgement regarding this, NIOS and private students are eligible for NEET.