Old English wadan "to go forward, proceed, move, stride, advance" (the modern sense perhaps represented in oferwaden "wade across"), from Proto-Germanic *wadanan (source also of Old Norse vaða, Danish vade, Old Frisian wada, Dutch waden, Old High German watan, German waten "to wade"), from PIE root *wadh- (2) "to go," found only in Germanic and Latin (source also of Latin vadere "to go," vadum "shoal, ford," vadare "to wade"). Italian guado, French gué "ford" are Germanic loan-words.
Specifically "walk into or through water" (or any substance which impedes the free motion of limbs) c. 1200. Originally a strong verb (past tense wod, past participle wad); weak since 16c. Figurative sense of "to go into" (action, battle, etc.) is recorded from late 14c. Related: Waded; wading.
Forbade to wade through slaughter to a throne,
And shut the gates of mercy on mankind,
[Gray, "Elegy Written in a Country Churchyard"]