How to force a database update

My client tried a new theme on his website and when he went back to the original theme the site broke. In an effort to get the site corrected I made a back up of the site in a dev/staging site. When I opened this staging site I was prompted to update the database and magically the staging site works correctly. I can’t see how to get the main site to update the database. How can I get it to do so? Or is there some way to copy the database from the staging site and replace the main site’s data with that info without breaking everything.

As always this is urgent so any quick help will be greatly appreciated!

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