What will happen if I was picked up on a bench warrant and can't make my court date?

If, on the court date, you are locked in a different jail, you may have that jail contact the court. If you are in the hospital, you may have the hospital contact the court. If you are dead, your relatives may have the county medical examiner contact the court. If those do not apply, then it might not be true that you "can't" make the court date. Instead, it looks like you are giving a higher priority to something else you would rather do than attend court. That usually does not work out well for the defendant. Judges consider your duty to the court as much more important than other ways of spending your time. That is especially true when a person appears on a bench warrant, which might be seen as showing that you blew off your duty to attend court last time. Repeated failures to attend are treated more harshly than a single failure to attend. You can try to get your lawyer to change the court date. That is not certain to work but it could be tried. Even better, you could actually attend court on the date you were ordered to attend. Good luck.