Try this five steps approach:

1. Examine your attitude - Start with the assumption that your coworker actually does care about doing a good job but that he’s as busy as you are.  It is possible that he actually doesn’t care, but that assumption serves no purpose other than to frustrate you.

2. Make sure that your coworker understands why it is important - “We have a sensitive situation to deal with on Friday with customer ABC, and one of the critical pieces is XYZ.  Could you please send me the information on it?”

3. Acknowledging how busy he is - “I know how busy you are…”  Empathy is a relationship builder.

4. Help him help you - “This information is critical to the resolution of the situation.  If there’s anything I can do to help you get me the information by Wednesday, please let me know.”

5. Express appreciation and make it personal - “Thanks in advance Michael, I really appreciate your help on this!”