Birmingham keeps Central Time, UTC−6, moving an hour forward from the second Sunday in March to the first Sunday in November. India is 10 hours 30 minutes ahead in summer and 11 hours 30 minutes ahead in winter.
That puts a 10:00-to-19:00 Indian office day almost entirely inside the Birmingham night. A reminder written at noon in Pune and sent at once lands at 01:30 in a patient's bedroom in summer. Schedule anything that is not urgent for the Birmingham morning: 09:00 Central is 19:30 IST from March to November and 20:30 IST the rest of the year.
Replies are the exception. When a Birmingham customer writes in their evening, it is morning in India, and the 24-hour window gives your team the whole Indian day to answer.