Since paste deposition or screen printing is not a critical delay time element in the process flow, I would not recommend screening the paste until the unit is ready for component placement. It will take much longer to place the components than it will to deposit the paste.
With water soluble paste, I would be concern with the amount of moisture in the assembly area, as this chemistry is susceptible to the absorption of moisture which could impact its ability to reflow. Excess moisture will cause the paste to spatter during reflow.
Secondly if the product is set aside prior to component placement the concern will be its tackiness, or its ability to hold and secure the component prior to reflow. I would not worry about the board thickness as I don't believe it has anything to do with the performance of the solder paste. There will be a large area of exposed solder paste which will dry out if the product is left out prior to component placement.
The bottom line is place the components on the board as soon as possible after the paste deposition process and reflow the board.