Well, what would the Sumerians do with a steam engine if they made it in the first place?
Good question. I can't see any immediate application that waterpower couldn't do better. Bear in mind, early steam engines (and you won't start out with a Watt-style pressure system, let alone a forced-draft or multiple-expansion design) are inefficient and cranky devices, big, slow, fuel-hungry and very much stationary. What could they use one for? Mills, bellows and perhaps fulling mills, flax processing and papermaking, if you give them that invention. None of that sounds like it makes the investment worthwhile.
OTL the first application was draining deep-shaft coal mines, which ensured a supply of cheap and plentiful fuel on the spot and did not require much by way of reliability (a mine doesn't flood if the drainage is out for a few hours).