As far as I understand their speed mostly comes from gravity assists from Jupiter, so you'd have to wait for suitable orbital alignments and the basic technology for "much faster speed" doesn't exist.
…but we’re very close. The next technique will be flying near the sun then deploying a solar sail for a huge speed boost. Voyager goes 3 AU/yr, solar sail boost with todays technology will enable 7-9 AU/yr.
Highly recommend watching Slava Turyshev discuss his work on an SGL telescope, which employs this technique.
https://www.youtube.com/live/lqzJewjZUkk?si=57VS4oqbaKEXmyOR
Various forms of nuclear propulsion have been investigated. None of them are anywhere near ready for missions, but that seems to be more due to lack of investment in their development (and environmental/legal/regulatory/geopolitical/etc concerns) than any scientific obstacle. If NASA/etc were really serious about it (as in willing to spend multiple billions a year on it), it could probably be made to work in only a few years.
The idea pursued nowadays is you launch using chemical propulsion and then only turn on the nuclear propulsion once you reach a safe distance from Earth. This is different from the original 1950s Project Orion which proposed to use nuclear pulse propulsion (i.e. repurposing nuclear weapons for propulsion) from the surface to orbit, which would have produced enough fallout to likely kill a handful of people per launch (in the long-run through higher cancer rates). The question then is - is it safe to launch nuclear material to orbit using chemical propulsion? Yes, we can secure it in containers designed to survive catastrophic loss of the launch vehicle. But, will the general public believe it is safe, even if it actually is? Possibly not-which is a political obstacle rather than a technical one.
The other issue is that nuclear propulsion systems can be too large/heavy to launch on a single chemical rocket, but you could launch them as multiple modules assembled together in orbit.
I don’t think this need or should depend on off-Earth manufacturing or mining capacity. I think it is going to be a long time before the highly complex manufacturing supply chains needed to turn raw materials into cutting edge technology like nuclear space propulsion systems exists off Earth. But we should be able to manufacture them modularly on Earth, such that in space we’d be doing module assembly rather than manufacturing.
I think that’s probably a very optimistic timescale. If there’s heavy industry on the moon inside 50 years I’d be surprised.