They used several Mars and earth gravity assists to raise Rosetta's aphelion to around 5 AU. The comet's aphelion is also in that ball park. The comets from Jupiter's neighborhood (about 5 AU) are moving about 40 km/s when they're in our neck of the woods (earth moves about 30 km/s). A comet moving 40 km/s wrt the sun at 1 AU can be moving anywhere from 70 to 10 km/s wrt to the earth, depending on inclination. A head on collision would be 30 + 40. If the comet was moving the same direction as the earth, it'd be 40 - 30.
A comet from the Kuiper Belt would be moving just a hair under escape velocity when in our neighborhood. Escape is sqrt(2) * circular velocity. So a comet from the outer system would be moving 42 km/s wrt to the sun. The velocities wrt to the earth could range from 72 to 12 km/s. The Kuiper Belt is 30 to 50 AU from the sun.
A comet from the Oort Cloud would also be moving a hair under escape velocity, 42 km/s in our neighborhood. But Aphelion would be much higher, up to 50,000 AU. Presumably many of the non-periodic comets are from the Oort. Click the comet links in this list and you can compare discovery date with last perihelion. For example Comet C/2011 W3 (Lovejoy) was discovered November 27, 2011 and it's .55 AU perihelion occurred December 16, 2011.
In JPL's description of the LINEAR NEO search program, they say "… with most of the efforts going into searching along the ecliptic plane where most NEOs would be expected. " But objects from the Oort don't tend to the ecliptic plane. In my opinion a high inclination comet from the Oort with our name might escape undetected until it was way too late to do anything about it. But an Oort comet with our name on it is very unlikely.