How long does it take for threadlocker to cure?

It really depends on which threadlocker you've used, the surface you've used it on, and the temperature of the materials. From this Permatex pdf document, it states the following for Blue threadlocker:

The rate of cure will depend on the material used. PERMATEX ® Medium Strength Threadlocker BLUE will react faster and stronger with Active Metals. However, Inactive Metals will require the use of an activator (Surface Prep) to obtain maximum strength and cure speed at room temperature.

Here is the associated chart with this:

It also depends on the temperature of the item. If the dry time is much too slow for you, you should be able to speed up the process with a heat gun:

Cure speed vs. temperature The rate of cure will depend on the ambient temperature. Full cure is attainable in 24 hours at room temperature, 22°C (72°F), or 1 hour at 93°C (200°F).

If you are using the product on an inactive metal (as shown in the chart above), you can use an activator to get it to cure. An activator can get the product to a "cured" state much quicker and much stronger than without:

There is other information given in the pdf which may be pertinent to your needs. If you aren't using Blue Threadlocker, I'd suggest the process is still the same, but the times may be a bit different.

As far as water goes, the Threadlocker should be contained between the threads of the bolts. It cures anaerobically (without air), which to me means, if used correctly, water shouldn't be getting on it and causing you any issues. I guess if the fastener is completely submerged in water it could be cause for concern, but once the product is fully cured there should be no issue. If time is of the essence, use heat and/or activator and you should be golden.