The literal reference is to a building that is absolutely, completely ready for occupancy. There is nothing the buyer needs to do except turn the key in the front door and move in.
In the context of a project or a contract, "turnkey basis" means an absolutely complete, finished job, beginning to end. The buyer doesn't need to prepare anything, know anything or do anything; all the buyer needs to do is pay. When it is over the buyer doesn't need to finish anything.
I'm a little puzzled as to what this would mean for an occupational health and safety consultancy. I imagine it means they can do this for a company that's never had it done before... and when they are finished, the company is completely ready to get a government inspection.