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An answer to the question of How well would the Moon protect the Earth from a Meteor? mentions as a possibility that the Moon could get knocked into the Earth.

What is the smallest change to the orbit of the Moon from being impacted by a large meteor that would cause it to eventually impact the Earth (i.e. "circling the drain")? What timeline would that look like (minutes, hours, days, years, etc)?

There is no possibility whatsoever of the moon getting knocked out of its orbit by an asteroid impact. Compared to the moon, even a large Chicxulub-type asteroid has a very tiny mass, and the moon has already been struck by several of them, but as you can see, it wasn't knocked out of its orbit. The largest asteroid in the asteroid belt is Ceres, 500 miles in diameter. Its mass is very small compared to the moon, but if a miracle caused it to leap out of its orbit in the asteroid belt, half way to Jupiter, and make a bee line for the moon, an impact at 25 km per sec might just be enough to produce a very slight wobble in the moon's orbit, but nowhere near enough to send it toward the Earth. The moon is actually moving away from us at the rate of several centimetres per year.

As several people have said, this is incredibly unlikely. Part of the reason why is that the "circling the drain" effect you describe doesn't really happen for solid objects much less dense than black holes. Orbits are not "precarious" in that way.

So, suppose something large enough and fast enough to change its velocity noticeably, but not large enough or fast enough to shatter it, did hit the Moon. The effect would be to shift the Moon from its present almost circular orbit around the Earth, into an elliptical one. Depending on the direction of the impact, it would either get a bit nearer to the Earth than it is now, once per orbit, or a bit further away (it also might swing North and South a bit). What is important though, is that this elliptical track is stable at least for a while. Suppose it gets knocked into an orbit that is 220000 miles from the Earth at its closest and 240000 miles at its furthest, that is where it will stay. It will not "spiral in".

Over a long enough period the gravity of the Sun also comes into play and things may shift a bit, but that is a relatively small effect.

Now, suppose that the impact was really big, or perhaps there were a long series of impacts (starting to look like enemy action..) so that the innermost point of the ellipse was eventually driven down to within a few thousand miles of the Earth, somehow miraculously not smashing the Moon to fragments in the process. At this distance it starts to matter that the near side of the Moon is closer to Earth than the far side, so that Earth's gravity pulls on it more strongly. If it orbited closer than about 3000km to the surface of the Earth for long (the Roche limit) these forces would eventually pull it to pieces, and Earth would probably have a pretty set of rings for a short time before internal collisions between the bits caused them to rain down on Earth and kill everyone.

Finally suppose the impact(s) was(were) so big that they actually put the Moon into an elliptical orbit whose innermost point was so close to Earth that the Earth and Moon touched. This is manifestly impossible without shattering the Moon, but in that case, the Moon would indeed hit the Earth. The time for the impact would be about 1/4 of the Moons current orbital period, which is to say about a week.

There are two issues at play here, only one of which is real.

It's possible to compute the energy and momentum that an asteroid impact would have to transfer to the Moon, assuming that two solid balls (classic Newtonian billiard balls) hit each other (either a direct impact or a glancing impact). There are certainly cases where the result would be the Moon going into an orbit which hits the Earth.

However long before the impact is big enough to seriously move a solid Moon, both bodies cease acting like solid masses and act more like drops of liquid. They splash, throwing both molten and solid rock into space in all directions at a variety of velocities.

In essence, this would be a smaller version of the events which are theorized to have formed the Moon in the first place, with a Mars-sized protoplanet (named Theia -- h/o/w/ t/h/e/y/ d/i/s/c/o/v/e/r/e/d/ i/t/s/ n/a/m/e/ i/ d/o/n/'/t /k/n/o/w) striking the very young Earth. See the Wikipedia article for a decent short description and pointers to more detail.

There are issues with this hypothesis as an explanation of the Moon's formation, but the broad outlines have been modeled in detail and are well-understood at this point. An impact big enough to seriously move a billiard ball Moon would release a very large amount of energy and throw a very large amount of rock into space in all directions.

Most of the loose rock would form a planetary ring around Earth before being captured by the remnants of the Moon. Enough would hit the Earth to be seriously troublesome. I haven't seen any estimates for a modern-day Lunar strike -- it's really way, way down on the list of things to worry about -- but back-of-the-envelope estimates make me strongly suspect that this would be a very good time to join Elon Musk's Martian colony...