I would say 'with the mechanic' or 'at the garage' My car should have been ready by yesterday, but it is still with the mechanic. My car should have been ready by yesterday, but it is still at the garage.

Should I use 'at latest' instead of 'by'?

"My car should have been ready by noon yesterday, but it is still at the mechanic." or "My car should have been ready yesterday, but it is still at the mechanic." I think are better.