I have done my Diploma in mechanical engineering. Am I eligible for pilot training?

Since you are doing your Diploma mechanical engineering, you would have definitely done maths and physics. You need to apply to the NDA for a job as a pilot in the defence. For being a commercial pilot, you can apply to a flying school, approved by DGCA.

An exam is held in all India for admission in pilot training. Its syllabus is almost same as 12th. So you don't have to worry about it. Your diploma degree is not required here.