Not literally in a sense that it must go in same direction as they went, but in distance to Earth.
I understand that this question may be a bit vague (is the budget 500M or 5B?) so feel free to assume: large budget: tens of billions.
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With reasonably current technology there are basically three options, I think:
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.
There is a 2011 MIT PhD thesis by Daniel B. White, Jr., that discusses nuclear reactors driving high-power electric propulsion devices. These propulsion devices seem like technologies that did not yet exist in 2011 but that aren't drastic leaps beyond the current state of the art. He models a list of missions, including one that he calls the Interstellar Precursor, which would fly to 250 AU from the sun in 10 years, with a 6.5 ton burnout mass and a very small payload. To get an idea of the scale of the effort required, he tosses around ideas like basically taking over the ISS in order to develop space-based reactors and propulsion systems.
With current state of technology if we wanted how long would it take for newly launched probe to overtake Voyagers?
So with considerable R&D and investment, it sounds like the answer might be 10 years, starting from a launch date maybe 10-20 years in the future. But with strictly off-the-shelf current technology, the flight time is probably much longer.