Wedding objections or the common phrase "speak now or forever hold your peace" is a Christian marriage ceremony tradition that was first introduced during medieval times. As you might imagine communication between different towns were not as easy as they are today, and to prevent secret marriages that were not legal, objections were introduced to allow all community members to bring forth any legal reasons why two people should not be wed - For example the groom to be being wed to someone else in another town!
As an Atlanta Wedding Officiant, I can say that I've been fortunate to not have had any objections during the ceremonies that I've performed. I should also note that these objections are not as common as they are portrayed in movies, as someone cannot object to a marriage because they're in love with the bride or groom. The objection has to have legal grounds.
Should someone object during the ceremony the practice is to pause the ceremony and speak to the objector privately to confirm the reason, which has to be legal such as the bride or groom to be being legally married to someone else, not being of legal age to wed, or if there is proof that either party are being illegally held against their will and forced into the marriage. Objections beyond that would usually be ignored and the ceremony would proceed as normal.
That being said, while I'm sure the presence of an unwelcome ex proclaiming their love would make for an awkward and possibly funny scene, it would be overlooked unless the bride or groom to be decided at the alter that they've changed their mind.