With reasonably current technology there are basically three options, I think:
- A Jupiter gravity assist similar to how the Voyagers themselves got most of their velocity. This could do a bit more by going closer to Jupiter (we know more about the environment there now, and our navigation is better); or by actively boosting at closest approach (how much you can do this depends on your budget and your payload mass). We can save some fuel for a boost at Jupiter by using multiple Earth and Venus flybys in the way that is now standard.
- A solar gravity well manouver: use Jupiter to drop us in close to the Sun and then use as much boost as we can possibly manage at closest approach. If you can manage to get to Jupiter with a big enough delta-V reserve, this is better than using it at Jupiter
- Long duration low thrust systems like ion engines. A nuclear powered ion drive vehicle with a big tank of xenon and few spare thrusters to switch to when the first ones run out could eventually build up a very high velocity.
In all cases, I think, the answer would be a few decades.
If you are really in a hurry you could consider the "nuclear shotgun", perhaps on the Moon. That could launch a reasonably large, very tough payload at a very high initial velocity.