To determine the correct answer, we need to analyze the validity of each statement based on current knowledge of the said programs.

1. Statement 1: "NISHTHA is an integrated teacher training programme that has been extended to cover teachers at all stages of school education." 
   Explanation: NISHTHA, which stands for National Initiative for School Heads' and Teachers' Holistic Advancement, is indeed a comprehensive program aimed at improving the teacher training landscape in India. It has been extended to cover teachers at all stages of school education, ensuring that educators at every level, from primary to higher secondary, receive necessary training. \(\Rightarrow\) The statement is correct.
2. Statement 2: "PM POSHAN covers only those school children who are studying in classes I-VI in Government and Government-aided schools." 
   Explanation: PM POSHAN (Pradhan Mantri Poshan Shakti Nirman) is an initiative that provides nutritious meals for school students. However, it extends to children studying in classes I to VIII, not just I to VI, as offered in government and government-aided schools. Therefore, the statement "covers only those school children who are studying in classes I-VI" is incorrect. \(\Rightarrow\) The statement is incorrect.

Based on the above analysis, only Statement 1 is correct. Thus, the correct answer is '1 only'.