I do not have any data. This is based on my personal observation over the last 4 months - during which time I visited Ramachandra Medical College Hospital, Apollo Hospitals, MV Diabetes Centre and Shankara Nethralaya in Chennai.
In each of these hospitals, I noticed a huge crowd, but what was interesting was the presence of patients who spoke Malayalam, Telugu, Hindi, Bengali and a few other assorted North Indian languages. In each Hospital, the staff were equipped to deal with the patients who could only speak in the above languages. The staff were all - or almost all - Tamilians but they spoke perfect Bengali, Hindi, Malayalam, Telugu and what not besides Tamil and English.
In Apollo where my father underwent a heart bypass surgery, I found NRIs from Malaysia and USA. I also found a few Africans and Caucasians (not sure of their nationalities). In and around Apollo Hospitals in Greams Lane, I found plenty of Malayali and Bengali patients staying in temporary accomodations.
I thought that the facilities for something as complex as heart bypass surgery are perhaps not widespread and perhaps Chennai has over the years built up sufficient number of private hospitals that specialise in heart surgery. But when I went to Shankara Nethralaya, I found several people who seem to have clearly come from North India and from Bengal for something as simple as an eye checkup and proper prescription of glasses. This is difficult for me to understand. Surely there should be enough eye hospitals around the country? It was strange to see the staff at Shankara Nethralaya talking to the patients in perfectly acceptable Bengali and Hindi. Unless the stream of visitors is large, the staff wouldn't have taken the effort to learn these languages.
So what is actually happening? Does anyone know?
As an aside, I wonder if people in North India will take the effort of learning Tamil or Telugu to talk to their customers.