There are very few private chain manufacturing specialist facilities, so no common term evolved. Chains are typically manufactured in much larger shops along with a variety of other products, generally only as they relate to one another in some end-product. Any terminology used by the staff of such a factory for "the place where we make chains", (such as "the chain shop", or "chainworks") would be shop slang. Unless/until such terminology became widespread across industry, or a selection of dedicated chain manufacturing subsidiaries began using common terminology for themselves, there is no one word for such a place. In general though, a production line for any given product will be called a "product"-mill, "product"-shop, or "product"-works, unless they refer to it by the material being consumed, or the process used. A modern chain shop could just as easily be called a steel mill or simply a "forge welder". If it's in the same building as other product lines, it might even just be called a "fabrication shop".
The main reason for this decentralization of chain manufacturing is that, while most chains are cosmetically identical, they are generally made with very specific properties to serve a specific intended function. Chains used for rigging of crane equipment are very different from those used as guarding, which are themselves different from those used as a mechanical drive, etc. You can't take a chain made for the guard rail on the side of a farm machine and use it to hoist something by a crane- they don't have the same properties. (Well, technically you can, it's just a bad idea.) As a result, chains are generally manufactured by the same people who make whatever it is meant for, even if they have subsidiaries or separate factories to carry out the process. Rigging chains are made by crane equipment manufacturers, towing chains are made by heavy equipment manufacturers, tethering chains are made by boat manufacturers, decorative chains are made by jewellers, etc.
Even though early smithies and foundries did not have a contemporary theoretical understanding of the chemical and physical properties of Steel, they did understand basic metallurgy and how to control the physical properties of their products. Ancient blacksmiths would have been well aware of the fact that a chain made for one thing probably wouldn't be good at much of anything else aside from simply being a chain. Because traditional chain smithing was a slow, difficult process requiring precision and skill, they would have only been made in shops which had specific need for them, or dealt with customers who had regular need for them. That trend simply carries on to this day. Being a specialist "chain maker" doesn't make much sense if each of your products are only technically (and legally) valid for a very limited range of functions and you have to compete with your customer's own production team.