As of yet, there's no known weapon system capable of shooting down asteroids in the minutes or hours before impact, because of their sheer velocity.
And exploding a large one too near - where most of the material still hits but as smaller objects - can be as bad or worse than a single impact. Not so big local and tsunami effects but potentially multiple localities and potentially major atmospheric heating. The Chicxulub impact is thought to have sent so much hot ejecta material around the world that forest fires on the other side of the world were started by the downwelling heat.
Early detection is crucial, obviously. For objects inside the solar system with Earth crossing orbits - which should be most of them - the potential is for finding them with a lot of time to act. For tracking there is the problem of enough precision - deflecting something that potentially could hit too early or with limited deflection capabilities risks sending something that would not have hit into a worse trajectory.
NASA thinks they have identified all the objects of planet killer size within the solar system, with none presenting danger. Tracking ever decreasing sized ones, big enough to be major disasters is a significant, long running undertaking - but it is the most public supported space agency goal. Way ahead of colonization or even crewed missions. I am sure other nations can and will contribute if their observatoreis already don't. A significant impact any time soon would be a powerful motivator - not that we should need to suffer a major disaster to know prevention will be worthwhile; foresight should be enough. A long way between tracking and having the ability to act of course.
Those with known orbits offer opportunity to match course, possibly on their passage around the sun, possibly using solar electric propulsion whilst going the same direction, to have time to dig bombs in deep for explosive dismantling or to change it's direction, or set up thrusters or try Gravity Tractors (keeping a spacecraft within it's gravity well, boosting at precisely the amount to stay at a fixed distance).
But there are the ones coming from further out, that aren't in known orbits, that won't give long warning. They will also offer less opportunity to make use of any solar propulsion or solar energy in the inner solar system and the delta-v for matching with it far enough out (rather than hitting it, shooting it ) can be very large.
Different types of objects and different orbits, different approaches. Different for something that crosses on the inward part of their orbit and one that would cross after passage around the sun on the outbound trajectory. Some, especially comet types, might be deflected simply by spraying black over them to promote outgassing - or white to reduce outgassing.
Or send industrial versions of those auto ball throwers and robots to load them - to toss material out to space.
But I think nothing will deliver more energy with less payload than nuclear weapons; having them used for good to prevent destruction would be different.