I recently purchased a flat packed cot for my soon to arrive baby. The wood is plain without any paint or varnish. How best should I finish off the wood on the cot in a way that is both durable and safe for the baby?
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.
If the baby is arriving in more than a month, then pretty much any Latex paint should be ok. Latex paint cures in 30 days or so.
If you're super concerned, you can find low VOC (volatile organic compound) paints online, although they're more expensive, and considering the number of kids that grow up in painted cribs w/o any harm, probably overkill.
I'd steer clear of oil based paints or polyurethane finishes unless you have a long enough time for the finish material to completely cure.