I suppose it had to happen sooner or later (i.e. the move to Excel 2016)!
From your last post, I assume that you are not receiving any error messages also.
This is a real head scratcher type scenario as I have heard many similar stories to yours. Its usually a compatibility issue and the same thing seems to happen when people use Excel for Macs. Its a hard one to sort out especially when a network is being used (and I assume that you are).
I'll have a look at it for you tomorrow but in the meantime, in the main spread sheet on each computer, click on the File button at the top left of the sheet, select "Options" and then select "Trust Center" in the window that appears. In the next window that appears, click on the Trust Center settings button and then select "Trusted Documents" in the following window.
Here, make sure that the "Allow documents on a network to be trusted" box is ticked (do this on each machine).
Another method (although tedious) is to actually re-write your preferred macro line by line EXACTLY
as it is now into the VBE of one machine that has 2016 installed, save it and then try it again. Once saved, go to the VBE (Visual Basic Editor) in the Developer tab (or just press Alt + F11 to take you there) select Tools--->References. A VBA References window will open. Here, make sure that the box beside Microsoft Office 16.0 Object Library is ticked and, if it is there, tick the box beside Microsoft Office 14.0 Object Library.
(14.0 is for Excel 2010, 15 is for 2013, 16 is for 2016).
Before closing the VBA References window, click on the save icon in the tool bar over to the left (little blue box shaped thing).
I'm also hoping that fellow Moderator ac3mark
will pay a visit here as this type of thing is "right up his alley" as a Master Programmer and his input could enlighten us all.
Let us know how it works out.