The bushings are already calling for a change on most cars, so your mechanic is right. That play in the steering that you're feeling is exactly what happens when they wear out: the bushings lose their ability to absorb movement and the components start to have more slack. The dangerous part is that it's not just a comfort issue - it directly affects how your car responds in curves and braking. If you put it off too long, the wear spreads to other suspension parts, and you end up paying a lot more.
My advice is to get it done relatively soon; it's not something you can put off indefinitely. It's not an emergency if the car still drives okay, but the longer you wait, the worse it gets. Plus, worn bushings can speed up tire wear, so do the math: by changing the bushings now you avoid replacing tires prematurely. If your trusted mechanic recommended it, he probably saw that they're really worn, not just a bit tired.