When trying to work out how long it might take for comet materials disturbed from the Oort cloud and Kuiper regions to reach the inner orbits and start affecting us in earnest, one might consider how long it took the Voyager spacecraft to get from Earth out to and beyond Pluto.
They were launched in 1977, and only very recently (like in early November of this year if I'm not mistaken) has Voyager 2 officially left the Solar System. That's about 42 years. -And it was traveling at 54,000km/h. -Which is 15,000 meters per second. For reference, a rifle bullet will travel between around 1000 and 1500 meters per second. So.., the Voyager spacecraft was moving about 10 x faster than a fast rifle bullet. And it took 42 years to reach the limits of the solar system.
So.., how long does it take before inbound comets reach us? It depends on their speed and starting time. How long ago did the C's say Nemesis was bowling through these debris fields? And how fast do comets fly? The rest is just math.
I just had the same question come to mind. It seems from what I've seen that the velocity of Asteroids and Comets can vary in their speed from 10km/s up to 600 km/s. As rightly pointed out though, that velocity is most likely not a constant for most if not all of those objects. It can vary greatly during their travel due to gravitational/electrical influences (like other planets), orbit and probably even more factors. Also, it seems that they (in general) accelerate as they come closer to the sun (the heaviest electrical/gravitational force in the solar system) and thus the closer they come to earth. So it is pretty much impossible to assign a correct time frame in which a hypothetical object (or a swarm of them) from the Oort cloud can reach earth when it gets pushed into the inner solar system by the "suns companion".
What we can do though is to estimate the fastest and lowest estimate if we would assume that the velocity of the object follows a straight line towards earth (which it isn't, since it is always an orbit or curved trajectory) and is always travelling at that speed (which it isn't either).
So going with 10km/s, we can calculate:
864,000 km ≙ 1 day
822,790,000,000 Kilometers [5500 AU] ≙ 952,303.240 days = 2609.049 years
So going with 600km/s we can calculate:
51,840,000 km ≙ 1 day
822,790,000,000 Kilometers [5500 AU] ≙ 15,871.720 days = 43.484 years
So we can say that the time it would take for such an object from the Oort cloud to reach earth can lay anywhere between at most 2609 years and at least 43 years. Go figure! If they can travel slower or faster, the estimates in both direction can increase even further, in both directions. So any number in between the two given above can hold true for any given object or swarms of them in terms of how long it takes for them to reach earth.