Depends where your are, chipper is common to some, others say 'fish shop', some 'chippy'. Other people say they're "getting chips", or "having carry-out" and leave the place to be entirely implicit.

That said, I've never seen it called the "fish and chippery", but it wouldn't surprise me to see that used.

And in some alternate universe, "fish and chips" became "Molly's Lips" in rhyming slang, leading to phrases like "get lunch at the molly".