I am traveling from NY to Mumbai via London, on H1B visa. If I am allowed to board at NYC, will I be sure I will not be troubled at London for immigration reasons? In other words with the British Airways agents do due diligence at JFK while checking my passport / visa's etc ?
Yes, airline staff in general will do a check that you have the necessary documentation to enter your destination country. This is because they are subject to costs if they fly a person who is not permitted to enter the country. So if you do not have the necessary documentation you will not be permitted to board the aircraft.
On the other hand, just because you have been allowed to board the aircraft does not mean that you will be allowed to enter the country. Airline staff are not border officers, and can't give you permission to enter. If you are on a security watch list, or your visa is a forgery, or the immigration official simply decides that your story does not add up, you can be excluded at the border.
You do go through a security check at LHR, but it is pretty perfunctory (by British, attack-dog standards). If you are traveling to India on an Indian passport, they won't ask any questions.