Could a Druid use a Wand of the War Mage as a Druidic focus, if the wand was made of wood?
Probably. A druidic focus is defined as:
A druidic focus might be a sprig of mistletoe or holly, a wand or scepter made of yew or another special wood, a staff drawn whole out of a living tree, or a totem object incorporating feathers, fur, bones, and teeth from sacred animals. A druid can use such an object as a spellcasting focus.
So, if the Wand of the War Mage is made from "yew or another special wood", then the Druid should be able to use it as a focus. The only requirement the Wand of the War Mage itself has is that it requires attunement by a spellcaster. A Druid is a spellcaster, so this isn't a problem. As long as it's made of the right wood, the Druid should be able to use it, and get its bonuses.