I am not a home student for fee purposes but I do not require a student visa. I am allowed to live and work in the UK. Is it possible for me to enroll in PhD as a part-time student?

It will be possible to enrol for a part time PhD, but funding will still be a problem. Once you have lived in the UK for 3 years for purposes other than education you will qualify for full research council funding. It may be worth trying to get a relevant job for a few years and then trying to get a full time funded PhD place.

Thanks for the answer.
Sole reason I am going for a part-time phd is funding.

If you are working for an academic within a university, then you can register for a part-time PhD and this will effectively mean that you are fully funded, as the fees will probably be waived. Unfortunately it can be very hard to find such a position.

I am faculty in a busniess school in India. How can I do a part time PhD????

If you wish to remain in India you will have to look for a distance learning programme. It is likely that you will have to pay your own course fees.