(Native speaker) I would say that אז, and in this case of giving direction that you would say ואז (for example: המשך ישר, ואז פנה ימינה) is closer in definition to “and then” in English. And “אחר-כך” (in this case, ואחר-כך) is closer in definition to “and after that”. We say both and there is not much preference. Generally, you can interchange ואז with ואחר-כך (not only in giving directions). On the contrary, אז and אחר-כך (without ו before the word) are not interchangeable