I found this question because I was facing the same issue. I believe I sorted out a fix.
If you rearrange the columns on your table view to be in the order you want them on the project pages tile widget, it should automatically update to reflect the new order.
Edit: Upon rereading the OP’s first comment, it looks like they tried rearranging columns without luck. I will note that I did this under the “Main Table” view with success, but when I tried under “My Work View”, it did not work, despite “My Work” being the default table.