That's... not what happened. England started experimenting with steam because it _could_. Newcomen's atmospheric engine required metalworking that was flat-out

That's... not what happened. England started experimenting with steam because it _could_. Newcomen's atmospheric engine required metalworking that was flat-out impossible before the advent of modern(ish) iron-making processes. Greeks would have needed to perfect large-scale iron-making. And in England's case, it was itself driven by the need to manufacture things like cannons, anchors, chains, etc. If you somehow got transported into Greece with a perfect knowledge of steam engines, you first would need to spend several lifetimes researching the iron metallurgy. You would be able to build a toy steam engine out of bronze, maybe, but good quality bulk bronze was expensive enough to be used as currency.