What is a durable, child-safe finish for a child's cot?
Most modern enamels (aka "oil paints" or "alkyd paints") cure fully in 14 days, at least according to the product data sheet provided for the Sherwin Williams paint I just used for my friend's kitchen cabinets. Either way, an oil or water based enamel will provide a much more durable and resilient finish than a latex will. 

I would suggest sanding the wood with 220 grit sandpaper, applying a high quality primer compatible with the final finish that you choose, and then a high quality water-based or oil-based enamel paint. I've really liked the results I've gotten with Sherwin Williams Interior Water-Based Alkyd and Benjamin Moore Advance Water-borne Enamel paints. They are true enamels, but clean up with water. Again, be sure to use a compatible primer. According to the attached technical data sheets, the waterborne alkyd paints take 30 days to reach their full hardness.