I coat the paper for cyanotype, pt/pd and gum under tungsten light (don't use fluorescent light - it can fog the paper) and hang to dry in the darkroom - on a clothesline, with clothes pegs. If I'm doing any kind of a production line, the first ones are dry before I'm done and I move on to printing. Those are never in the dark while drying. If I have dry sheets I'm not using right away, they are stored in a dark drawer and used as soon as possible. I've never had to press the paper flat before using, but I have ironed fabric impregnated with cyanotype chemicals before using.
I wanted more room for my recent batch of cyanotype Xmas cards, and set up in the kitchen for 2 days; working in daylight, then tungsten as the light faded. The weather forecast was for rain all day, but the sun did come out in the afternoon. My kitchen is bright, windows on 3 sides, and there were beams of light right across my just coated paper. They were fine.
Relax and have fun.
In case it's significant - I use traditional formula, not the Ware formula.