Slowing or speeding the object will by definition change it's flight path
Slowing will make it fall to a lower/faster orbit closer to the sun
Speeding it will push it to a higher orbit farther from the sun
The issue with doing either is that it keeps the object in the plane of the ecliptic (the orbital plane of the solar system) where it can still potentially
interact with another astronomical body and have it's orbital path altered again...
Pushing it up or down and giving it an orbital inclination relative to the solar system can place it in a path where it can potentially never again
interact gravitationally with another body, so it will tend to STAY in the "Safe Place" where you put it.
Leaving it in the plane of the ecliptic does not eliminate the risk of future orbital path changes due to perturbations, unless you degrade the orbit so far as to put it on an intentional collision course with another planetary body or attempt to plunge it into the sun (it would likely melt and vaporise before it got there)
Ideally you'd want to detonate the nuclear weapon, in this case "nuclear propulsion charge" as near to the axis of rotation as possible.
Basically at one of the asteroid's "poles"
A nuclear bomb detinated CLOSE to an asteroid IS a "propulsion device"
But it does not require landing on the asteroid it's only got to function correctly ONCE and for about 9 nanoseconds to do it's job
Instrumentation of asteriods, radio transponders to accurately track their orbital path to a far more precise degree is logistically no less difficult as visiting each one that is known to be a potential threat with a single 5-10megaton device exploded next to one of it's poles and increasing it's orbital inclination and shoving it out of any potential interaction with any planetary body.
The object of "moving" something as heavy as an asteroid is to move it as little as possible while decreasing it's threat as much as possible.
"out of the plane of the ecliptic" is the best way of accomplishing this.
The other thing to be remembers is the farther it is from a potential impact the less vector needs to be applied to accomplish the desired effect, so moving it decades or years in advance is better than months.
If only for the fact that if the brutal but effective method I'm on favor of fails you have a long of time to try again... and again... as often as necissary to achieve the desired result.
Instead of one large nuclear device a series of them arriving a day or more apart
Look up the original concept of an "Orion spacecraft", this is the vary basic "bomb and iron plate" concept, in vacuum the bomb turns a thin layer of the plate into incandescently hot gas that through simple neutonian physics drives the plate away from the detonation.
Hollywood is always enamored of the "it looks really Kewl" special effects or
the "Tennis ball covered with burning tar" meteor...
Anyone who has ever seen a real meteor knows what it looks like, a bright white point of burning light...
a 200foot wide nickel-iron meteor like that which created "Meteor crater" near Winslow Arizona,
if you were close enough to that impact to see the initial fire trail you were vaporized by the
impact blast. Anything far enough away to have survived would have seen something indistinguishable
from a nuclear test.
as for "Space explosions" in most cases of you are close enough to be worried about shock waves
there was probably enough for the first burst of X-ray and gamma radiation to have turned you off
like god cut the puppet strings long before the fast neutrons arrived.
And those neutrons would be flying through a diffuse, ionized gas cloud that used to be you,
because the bulk of the x-rays turned you into incandescent gas.
It's kinda like a physics trivia question if the sun suddenly changed color (or went black) while
you were watching what would the visual effect be like?
And the answer is kinda weird until you think it through...
The disc would NOT turn a different color all at once, it would change color in the center
of the disc and the color change would rapidly expand to the edge taking just over two seconds.
Now think about WHY....
the Edge of the solar disc is just over 2 light seconds further away from your eye than
the center of the solar disc.... Of course if the sun did suddenly change color (or go completely black)
even if you were watching intently you wouldn't know about it until eight minutes after it happened.
Because the earth's nominal distance of 93million miles from the sun is also about eight light minutes away.