I have sat in on presentations from astronauts and an astronomer who work at NASA on this very topic. He’s a buddy of mine, who was the guy talking to the White House about Chelyabinsk. Unfortunately, I don’t recall details of numbers and sizes. However, it is a topic of interest.
The gravity tractor approach has the problem that first you have to build a fairly large vehicle (like multiples of the ISS in size/mass), then you have to move that vehicle into the path of the asteroid in question somewhere out there in space way away from Earth, and then you have to have time for the gravity tractor to work. The larger the object, the longer it takes and the more mass you need in your tractor. Using some of the asteroid itself is an idea to increase the mass “on site”, but that has complications of its own. Suffice it to say we wouldn’t be able to do this anytime soon.
The stand off nuke approach actually is the best option for most cases. It doesn’t rely on the object having structural integrity so you can push against one side of it and the force will transfer through. Many of these objects are “rubble piles”, aggregates of rock. The gravitational force holding them together is tiny. Astronaut Stanley Love built a table top demonstrator - he suspended rocks from string in a framework so they hang next to each other. Push on a rock, and the others just slide out of the way. And he says that is something like 10 times stronger than the gravity in question.
The stand off nuke also works for objects of a larger size than anything we could reasonably try to affect with a gravity tractor, and the time scale to cause the effect goes down. Of course, for best effect, you want a series of stand off nuke blasts.
There are basically two options for changing the path. You don’t want to try nudging sideways. That takes far more energy, and on rotating bodies is going to have even more complications. Rather, you want to either slow it down or speed it up. That means either blasting in front of the path or behind the path. The idea is to put the asteroid through the Earth orbit window in a different time than when Earth is hitting that spot.
Then you get the tricky politics of do you speed up or slow down, what does that do to the impact point, does it move it into another country? The sensible answer is to move it the least to the most desolate area, but nobody is going to want you to move the asteroid impact into their country in case the result is not completed or off target. They will want it moved away from them. If everyone wants it moved away from them, there’s nowhere to move it.
Right now, the emphasis is on (a) identifying as many NEOs as possible down to some “reasonable” size, and (b) characterizing the asteroids, their structures, and what happens when you try to do things with them.
(a) is complicated by the very few number of people actually working on it. It was described as currently there are about as many professional astronomers working on this as the staff of a moderate-sized McDonalds. That may have doubled to 2 McDonalds by now. It’s also complicated by defining “reasonable size”. The smaller they are, the harder to find. But even 20 m objects are dangerous in the sense they can cause damage and loss of life - witness Chelyabinsk. Right now, I think “reasonable size” has been stated at 50 m.
(b) is where robotic missions to comets and asteroids like Hayabusa come in. It’s also where the proposed trip to an asteroid would fit. The modification to use a robot to drag a small asteroid to Earth orbit is less useful from that standpoint - the method of trying to grab it with a bag will do messy things to the structure. And it’s not particularly useful for testing methods for moving larger asteroids.
I would estimate that even if we had no political or financial hurdles, an we had an existing launch vehicle and space delivery vehicle to get the warheads to wherever, it would take a minimum of 2 years to be able to do something useful against an asteroid that would be a significant climate change event. YMM definitely vary.