The position for toys specifically is governed by the Toys (Safety) Regulations. This applies to any person who, in the course of a trade or business, supply or manufacture toys (this does not necessarily mean that the person's business has to be of actually supplying toys). With respect paints and protective coatings it sets out the maximum daily intake levels for nasty elements (Antimony, Arsenic, Barium, Cadmium, Chromium, Lead, Mercury and Selenium). Obviously, the key concern related to quantities of lead in paints (not an issue with new paints these days). As to your toy box, it may not itself constitute a "toy" for the purposes of the Regulations. You may not be subject to the Regulations if, for example, you are making the item for a friend. If you were selling the item commercially then the various bits of consumer protection legislation would apply (e.g. fit for purpose and generally not a hazard). I would think that most paints/varnishes are child safe once dry. If in doubt, I'd check with the manufacturers who I think have to provide hazard data for their products in any event.
Regards, Edmund