In Schmidt & Dilcher 2007, the authors were interested specifically on how amber can trap aquatic organism, so this is a specific, out-of-the-norm case study but it might give you at least on order of magnitude. Here is an extract:

Within the first day or two after initial contact with water, all three resin fractions were traps for limnetic organisms. The resinous film at the surface then dried out and broke into small centimeter-sized fragments, so no long-term preservation can be expected for this exudate. The pieces of resin hanging at the water surface became more and more solid within a few days because of air contact at their upper surface. After 1 week, these pieces were nearly solidified (Fig. 1 D).

And later:

In contrast, the large resin bodies on the ground did not solidify as long as they were covered with water. But this resin was also initially a trap for microorganisms before a thin hardened skin developed at the resin surface after 1 or 2 days in the water, which prevented tiny organisms from getting stuck. Nonetheless, the resin remained liquid inside, and therefore large arthropods that were able to break through this skin could become entrapped over a couple of weeks. [...] The resin solidified when exposed to the air in the laboratory or by decreasing water level in the swamp during the summer.

So the polymerization process to go from resin to solidified resin (copal) takes weeks. At that point, the system is already closed meaning the contained organisms correspond to that time interval.

The next step (amberification) consists of further polymerization as well as oxydation and can take several million years (Poinar 1992). However the minimum time needed for this process is unknown.

_{Reference:
Poinar Jr, 1992. Life in amber. Stanford University Press.
Schmidt A. R. & Dilcher D. L., 2007. Aquatic organisms as amber inclusions and examples from a modern swamp forest. PNAS, 104(42): 16581-16585.}