The short answer is that it is irrelevant. There is a much longer answer with at least three different reasons as to why it is irrelevant, but I can't be bothered to type it out now.
In British India, those living in British ruled provinces were British subject, while those who lived in princely states (the yellow parts of the map) were British Protected Persons (BPPs). So, for a start, it would matter if your wife's grandfather was born in a British province or in a princely state.
Then, after 1948, those British subjects in the Dominions of India and Pakistan who did not become Indian or Pakistani citizens after independence remained British subjects but without CUKC status. In modern parlance, British subjects without citizenship.
Your father-in-law's passport states that he is a "British Subject, Citizen of the United Kingdom and Colonies". That means that he acquired CUKC status.
Assuming that his father (your wife's paternal grandfather) was not born in a British colony (legally speaking, India was not a colony.), the only way your father-in-law would have acquired CUKC status would have been by naturalisation or registration.
Did you check both naturalisation and registration records with the National Archives?
Also check your mother-in-law's records. They may lead you to your father-in-law's records.
At this point, I am strongly inclined to suggest that your best option would be to move back to the UK en famille and register your children as British citizens after three consecutive years in the UK.