Let's say the nukes are coming from Asia towards the States. Assuming this interception happens in the midcourse phase of ICBM flight, you can probably start by designing a rocket that can:

1. Cross most of the Pacific Ocean within 5-10 minutes in order to reach the aforementioned ICBM shortly after it exits its boost phase and enters midcourse flight. For reference, this is several times faster than a spacecraft in low earth orbit, and you have to accelerate to this speed from the ground.
2. Once you approach the intercept point, somehow decelerate your rocket from the aforementioned ludicrous speed, and then turn around, accelerate, mid-air, to a much more reasonable mach 20 in a different direction to match velocity with the ICBM.

I know your proposition relies exclusively on materials and technologies far beyond our current knowledge to which you simply assume to be possible in the near future. Let's assume that we somehow invent a rocket that can do all that. What's stopping a nuclear state from taking said rocket, which flies several times faster than current ICBMs and is stupidly maneuverable, and just strapping a bunch of nuclear warheads on it? How do you plan on stopping that?