Quite, plus information can only travel so fast in a pre-digital civilization. As far as I'm aware, post is carried by man or bird in Thedas.
So Tevinter probably wasn't even aware there was a Blight until the Archdemon was long dead and Fereldan had already began to really recover.
Complicated answer: Asunder revealed that the mages have a method of fast messaging between Circle towers. So apart from normal means, the Fereldan Circle may have relied the message to other Circles across Thedas, probably at the Wardens' request. Despite the bad relationships between Tevinter and other Chantry countries, I'm sure the Imperium has Warden garrisons too. Then you only need prisoners, sailors, merchants or diplomats to make the tale reach Qunari ears.
Short answer: we know for sure that the Qunari and other major powers have agents and spies across the world, like in real life. The moment unusual darkspawn activity was spotted in Ferelden and the local Grey Wardens started warning about a Blight, I'm sure the fastest messengers were sent to the higher-ups in Minrathous and Par Vollen.