Old English wudu, earlier widu "tree, trees collectively, forest, grove; the substance of which trees are made," from Proto-Germanic *widu- (source also of Old Norse viðr, Danish and Swedish ved "tree, wood," Old High German witu "wood"), from PIE *widhu- "tree, wood" (source also of Welsh gwydd "trees," Gaelic fiodh- "wood, timber," Old Irish fid "tree, wood"). Out of the woods "safe" is from 1792.
parasitic insect infesting human hair and skin, Old English lus, from Proto-Germanic *lus (source also of Old Norse lus, Middle Dutch luus, Dutch luis, Old High German lus, German Laus), from PIE *lus- "louse" (source also of Welsh lleuen "louse").
The meaning "obnoxious person" is from 1630s. The plural lice (Old English lys) shows effects of i-mutation. Grose ["Dictionary of the Vulgar Tongue," 1785] has louse ladder "A stitch fallen in a stocking."
<a href="https://www.etymonline.com/word/woodlouse">Etymology of woodlouse by etymonline</a>
Harper, D. (n.d.). Etymology of woodlouse. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/woodlouse