I was once invited by an Irish musician to hear his band performing at a local teather on Tursday. Thyme/ time has been a pun since at least the Middle English period, so I suspect it had always been a hard T, the spelling etymologised by pedants in the 18th century, as happened with 'debt' and several other words. The Greek indeed has a theta, irrelevantly, and anyway the consensus is that in the Classical period, that would have been pronounced as t'h rather than th.

I like the way foreign spelling trips us up- the number of people who talk about Italian dishes containing funjee, when the Eyteyes themselves went to the trouble to put the H in funghi, to insulate the G from the I.