People who say they are busy do it out of habit and a feeling that's what everyone wants to hear. Yes I know we're all busy. Busy doing what exactly!? Chasing things which shouldn't need to be chased - like timesheets? Busy being incompetent doing things the slowest way possible, when they could have automated the task long ago for want of some direction from management. Busy waiting on other teams to tell you they are busy. And busy attending meetings which get cancelled because other people are too busy to attend.
If you want people to change then value what they do and reward them for doing it. And also just don't pay people who don't submit their timesheet. They'll soon start remembering to do it I'm quite sure.