If you spot an injured bird, you should preferably consider dropping the animal off yourself. For transporting birds, place them in a cardboard box and make some holes in the box for ventilation.
You can contact the following places.
Wildlife Rescue: Contact them preferably for dehydration based injuries or cut wings requiring stitches.
Where: Chawri Bazaar, Old Delhi
Contact: +919810029698, +919810129698, +919810639698
Jain Charity Bird Hospital: Following the tenets of Jainism, they treat sick and injured birds and then set them free, they have a hospital and Delhi and Gurgaon.
Where: Shri Digamber Jain Lal Mandir Ji, Opposite Red Fort, Chandni Chowk and near Govt Girls School, St Jain Mandir in Gurgaon.
Contact: +91 9810016162, +91 9968855520, +91 9213922978, +91 9810387093
Sansthanam Abhay Daanam, Charitable Bird Hospital