The 4.5 m tall concrete structure at Hawkeswood Metal Recycling's plant gave way on 7 July 2016 crushing the workers, before tonnes of scrap metal behind fell on top.
The five men were pronounced dead at the scene, while a sixth escaped from the debris with only a broken leg.
The HSE has taken over from the West Midlands Police and will investigate whether any safety regulations were breached.
Geoffrey Brown, the principal inspector who is leading the investigation, said: "This remains a criminal investigation into the five deaths. We will be considering whether there were any breaches of health and safety law relevant to the circumstances of this tragic event.
"We will of course keep the families updated as our investigation continues."