transistion between wards’ primary

It is difficult to have two units overlapping.  It sounds as if both groups are dealing with frustrating circumstances.

If circumstances around you aren't going to change (like stopping the time overlap), you will have to adjust your expectations or schedule to work with your goals for your Primary students.  Since you have an expectation that primary children clean up, and that expectation causes traffic issues with the other unit, it seems reasonable to include that as part of your closing exercises.  In other words, the closing prayer is given at 11:55am, allowing 5 minutes to clean up. During that time, students are not released to parents, but they are helping you clean and tidy up.

Can the placement of items in cabinets be moved around so that you have columns of storage instead of rows?

Can you ask each teacher to choose a class president who cleans up their row when class is over each day?  Can older children move the smaller chairs?  What other ways can you use the children in Primary to tidy up the messes they make?  Many hands make light work, right?

In the grand scheme of things, this is a minor annoyance. Be very cautious that you don't allow your frustration to affect your relationship with other servants in Christ.