I followed the procedure said in a XDA forum (the very first post) and I got the LTE working in my phone.
I confirmed the same and added a confirmation to that thread as:
I am using Nexus 5 (D820) purchased in USA and being used in Chennai
on Airtel network. I read through many resources about the LTE bands
that the phones (D820 and D821) support, about the bands that
different service providers support. After reading through the posts
in this thread I was aware band 3 can be enabled on D820. Even at this
moment of writing (after enabling the band 3 on D820) I believe from
various links Airtel uses band 40 in Chennai.
To my surprise when I enabled band 3 (after few tensed moments) I saw
LTE indicator in my mobile. The screen shot is as below:
The app LTE Discovery did report that my mobile is in LTE when I made
it to work in "LTE mode only". However it does not report the band
that is being used.
I tried Speed test, but it showed a disappointing speed as good as 2G
(1.5mbps download). I need to try this at other places and see whether
really 4G is enabled in my phone or not.
Edit: The download speed in other areas of Chennai are in par with 3G. However surprisingly, I could see the upload speed is as good as download speed in 4G.
In the following image, the middle entry is 2G and rest are 4G.
Also I used Nexus 5 Field Test Mode app and it says I am using band 3 - surprise.
Edit: This was performed in my Nexus 5 D820 running stock Android 6.0.1, stock kernel, rooted and with custom recovery TWRP.
**Edit2:**This change seems to be permanent. I had OTA updates installed and after a point of time reinstalled latest ROM and even tried locking boot loader. In spite of all these changes still LTE is working in my phone.