I'm tying to find any clue about the timescale of the biological pump. That the carbon is recycled to the surface ca. 1,000 years is clear but the timescale of the biological pump seems hard to find. There is an report I saw when I was surfing the internet saying that the timescale can range from 45 days to 5 years which I find is a little shocking. Does it truly vary that much, or there is something wrong about the data?
I'm going to answer assuming you are referring to the time between the moment when the organism absorbs the carbon and the moment when it is trapped in sediments. I'm also going to answer by talking about what I know best, i. e. diatoms, which happens to be the major component of the biological pump.
The variation you are referring to is consistent with the fact that coastal diatoms only have a few hundred meters to sink before reaching the seafloor, whereas open-ocean diatoms are sinking for ca. 1000 to 5000 meters. According to a compilation by Lisitzin (1972), so arguably a little dated, the settling velocity of marine diatoms in the water column is only of a few meters per day (he advances the number of 0.2 to 1.2m/day for 20 to 50µm-sized marine diatoms). But it can vary according to many factors and some of the coarsest, heaviest forms (such as resting spores) or diatoms that travel down in fecal pellets of copepods can be faster and reach the seafloor in an open ocean setting in only a few hundred days.
We don't usually consider this as being part of the biological pump sensu stricto but of course most of the organic matters created by diatoms from atmospheric CO2 is recycled by the zooplankton that grazes on them, which is in turn recycled by what feeds on that zooplankton, etc. Eventually some of them will end up trapped in sediments as well. This lengthens the process of course.
Lisitzin, A. P., 1972. Sedimentation in the world ocean. Society of Economic Paleontologists and Mineralogists, Special Publication 17: 218pp.