In the UK from a commercial flying point of view, ATC will quite often give the instruction for initial departure i.e. "after passing 500ft when ready turn L/R to beacon/point"
If no instructions are given we simply fly runway heading until the company acceleration altitude and then turn toward our first waypoint. Obviously some airfields may have high terrain or obstacles preventing a direct turn in which case following the first part of the engine fail proc until above MSA or turning and passing through the overhead is normally used.
There is no hard and fast rule, all depends on the airport and local terrain. Some local knowledge helps no end, especially if IMC or night.