The correct option is
C His
'Nitin' is the name of a man: "He was already late."
Option D: 'His' is a possessive pronoun used for men. ('His' modifies the noun 'way'.) It completes the sentence in meaning and hence option D is correct.
Option A: 'Its' replaces objects that have been mentioned before. But, 'Nitin' is a person.
Option B: 'Their' is a plural pronoun whereas 'he', 'Nitin' is singular.
Option C: 'Her' refers to a woman whereas Nitin is a man.
Hence options A, B and C are incorrect.