Start listening at about 24 minutes if you are interested in what happened on the ATC frequency(Director):
ATC recording
Shenzhen 9041 called ATC and advised a go around (may have been for separation, but that is speculative). Later, in the climb to 5000, they read back another aircraft's(Saudia986) clearance to "Turn right heading 350" as "Right heading 340, Shenzhen 9041".
ATC didn't catch the mistake and then handed them off to another frequency.
Yes, a right turn instruction from ATC should probably have been queried by crew, but they had been climbing for about 45 seconds in a go around. Hard to tell what it looked like in the cockpit for that turn. But they were probably well above the terrain and climbing, so it might not even have looked very exciting. Doubt the TAWS/ground prox was yelling, but it may have been when they were briefly pointed at Lantau Peak.
As usual, the news reports and comments on the web are not really based on much fact.
Crew made a mistaken read back. ATC didn't catch it.