I’ve never heard of this “thau” thing. I can’t seem to trace it beyond a really dubious 17th-century source, De dis Germanis, by one Schedius. He writes: Et sic unicum deum Thau Celtæ prisci colebant; & tres Personas S. S. Trinitatis, Hesum Deum patrem, Belenum Deum filium, & Tharamin Deum Spiritum Sanctum in uno Thau, id est, Deo venerabantur. (from the 1728 edition, on google books.) Even without wrestling with the Latin, it’s pretty clear that he’s trying to map the Celtic deities Esus, Belenos, and Taranis onto the Christian trinity. “Thau” must be the Greek letter “Tau,” that is, the letter that looks suspiciously like a cross. I don’t know where he gets that: it’s not a Celtic theonym.
The Shamrock is not the clover, as the Victorian reading would have it, but the sorrel. That doesn’t really help much with whether it meant anything to the pre-Christian Celts. It probably did; there seems to have been a rich symbolic reading of the natural world, but what it represented is not clearly outlined in any early sources.