I applied for a UK direct airside visa at New York VFS on 17th Oct 2018, as I am flying from San Diego to India via London. It is 30 Nov now, and I still didn't get back my passport. My flight is on 5th Dec. I cannot miss the date, as I have booked another connecting flight, as well my appointment in India to stamp the US visa. What are the options available for me? Any help/suggestion will be appreciated.
As written in a comment by @Giorgio:
If you're unable to get a response from the VFS centre through which you applied, your other option would be to contact the UKVI help centre: it is fee based in England and it available 24 hours a day Monday through Friday (so not from Saturday12 am through Sunday 12 pm GMT.)
At this point, calling would be faster than email (you pay for that). Also consider getting an emergency certificate from your government; it would not solve the UK transit, you'd have to rebook a direct flight from the US to India.
After you have done that, please check that you really need a visa to transfer in the UK, often it is not needed.
This question deals with that.