A possible process to deflect a comet or asteroid
A Rocket engine
A Rocket engine is no more than a tube, a plug at one end and a fast burning fuel.
A possible way to deflect an astral body
A comet might be easier than a metal asteroid as the initial process might take a longer time to complete than the time to impact.
First issue: Who and what needs to be involved
The first issue is to arrive on the astral body with the necessary equipment in time to affect a resolution. To do this it will be necessary to have the equipment and vehicles located in space at several space stations around the earth. To do this it will be necessary for every nation to participate in this endeavor, as it will take the whole of the earth resources to setup and man the station.
Second Issue: When and how will this work
This must be done immediately, waiting only puts us in a tighter time constraint. Once the station has been setup in space, it will be necessary to have this station manned at all times with observers, flight crews and the men that will be doing the actual work on the astral body.
Final issue: How to deflect the Astral Body
If a large tunnel/hole is cut to the approximate center of the astral body and it has been packed with spent fuel rods and an initial igniter the center of the astral body could be heated up enough to melt the center of the astral body. When enough heat is created, a Rocket Engine Action would take effect blowing out the tunnel/hole that was dug. The rocket engine effect would not destroy the asteroid but could give it enough of a nudge to move its trajectory