This has started for me as well around the time the background download for the next patch was released (I’m fairly certain it’s related).

However I, just now, seem to have resolved it for myself. After some digging I found a post on reddit about people being able to launch the game after only canceling the pop-up or connecting to a different region, canceling the pop-up, then going back to the original region.

Doing the steps above, going back to the Americas region (my original region) the grey arrow icon indicating background streaming download was appearing. So I clicked play, had the same pop-up, closed that window, but left the Blizzard launcher alone for 5ish minutes and HotS launched properly.

So try this:

Connect to a different region and try to launch the game;

Click Cancel on the “connecting to streaming server” pop-up;

Connect to your original region;

Leave it be for several minutes (5-10+).

Not guaranteed to work, but the root of the problem might be with the time it takes to establish the connection, rather than being unable to establish it in the first place.