The best time to cut saplings is in winter (December, January) when they contain a lot less sap. You can fashion a staff straight away from a winter-cut sapling.
And it does have to be a sapling. Heartwood is no good because it's too stiff. A quarterstaff needs some spring in it, which is why really stiff woods like sycamore are no use.
Avoid saplings with large knots, heavily decayed branches or fissures in the bark- these often signal weakness in the wood.
The best British woods for staffs are oak, hawthorn and blackthorn. Ash is OK but does tend to split and for the surface to flake off. Hazel is fine for a beginners staff but warps a lot and will break if hit hard. Softwoods (pine, fir, spruce) are too light and will also break. We don't really know about foreign woods but reckon that for North America, hickory would be good because it's used for making axe handles.