Learning how to take care of a baby is stressful enough. Doing this while in a foreign country, and while having to navigate bureaucracy to leave quickly, can be even more stressful. Fortunately, you have some options. You can get baby sitters reasonably cheaply to help care for the baby during the day and during the evening.
Hirinandani Hospital: If you are staying at the hospital, you can ask at the nursing station if they have sisters (nurses) that can stay with the baby either during the evening or during the day. The hospital will bill you directly, about 600 Rupees for a 12 hour stint. The sisters (nurses) are often the same nurses that work in the hospital, although sometimes the come from outside. Either way, they are trained nurses. They generally do not take tips. Night nurses are not always available; availability seems better on weekends than week nights.
Mumbai - Asha: Although there are certainly other nurses, Asha has worked with many couples, helping to care for their babies. She's been to the Embassies and the FRRO, so she has unique knowledge of both taking care of babies and getting through the surrogacy process. She really likes babies and tries to collect pictures of all the babies she's helped care for. Ask her, and she'll show them to you. Asha charges 2,000 Rupees (US$40) per 12 hours. She can be reached at ashamj70 at gmail dot com or +91 97657 45028 or on Facebook (phone is probably the best).
Know of other options? Please add them in the comments below.