In reply to Willi Crater:
^ This.
Transit time (from eating to pooing) in dogs is usually about 24-48 hours, but this is highly variable, so you need to be on scooping and dissection duty fairly straight away. Remember, it's only poo.
Things that can help to promote clearance include high residue foods - chappie is the usual favourite - along with a small amount of liquid paraffin. But provided your wife wears fairly discrete earrings - ie, no dream catchers or innately pointy jobs - then your hardest task is going to be persuading her to wear it again, once it reappears.
One note of caution - if your dog starts to vomit, goes off food or shows any sign of distress / pain / 1000-yard stare (you'll see it if it happens), seek veterinary advice immediately. It's unlikely with a small foreign body like this, but don't wait out any of these symptoms.
Sure it'll be fine - and make a good pub story.