I would say a decade of warning minimum. Getting any space mission off the ground takes years, and you want your deflection event to be as far out as possible. The flight time from launch to impact is also probably around a year on top as well.

But the good thing is that we don't necessarily need to scale up in size for a bigger rock. One impactor of a size we can launch on a current rocket can't get it done? No problem, send 10, or 100. The cost of the craft is going to be negligible on these, all it needs to have is a chunk of mass, some avionics, and an engine for course adjustments. All of these already exist, so you don't need to invent anything new. The biggest cost item is going to be the launch vehicles.