All of the answers have some merit:
In UK english 'forge' is commonly used for the entire building as well as the specific piece of equipment where metal is heated for forging. Similarly the actual heating apparatus may be called a 'forge' 'hearth' or 'fire' (when lit).
It is very common to see buildings in the UK called 'The Old forge', 'Forge Cottage' etc. Similarly working blacksmiths premises and businesses will often be called Something Forge.
Smithy is also correct but much less commonly used, an hardly ever for contemporary working forges.
As an aside it is also common to see the words 'forge' and 'foundry' confused. Forging involves shaping metal (usually hot) by striking with a hammer or shaped die whereas founding is casting molten metal into moulds.
Where we are talking specifically about producing weapons and armour 'armoury' and 'arsenal' may be appropriate but this tends to be more archaic for actual production of weapons and in modern usage implies more a place where weapons are stored or maintained rather than made.
Also an individual or company specialising in the production of blades could be called a cutler ( famously Wilkinson Sword) . Bearing in mind that historically the manufacture of the blades themselves would often have been separate from the hilt and other 'furniture' which were separate crafts.
This comes from my own experience of working in the blacksmithing and blade making industry in the UK.