Well, fairies are magic. If a fairy wanted to wield a diamond sword, they could and no audience would question it. Magic is the solution to a lot of problems. The weakness of a weapon forged with magic is, of course, a steel blade but that only matters if they’re encountering humans wielding steel on the regular. Fairies can do whatever they want and dance merrily on the graves of scientists the world over, so don’t let that stop you.
Blades of pure light.
Blades of diamond.
Blades from plants.
Fairies wielding magma blades or swords forged from stardust.
A sword of glass containing the beating heart and heat of the sun.
Futuristic fairies who behave like aliens in Iron Man style power armor formed from plastics/polymers wielding lightsabers and firing bolts of plasma.
They’re fairies. Sky’s the limit here. Except, it’s not because then we catapult ourselves out into space. Go however far your imagination takes you.
Look to myth for your solutions, especially the Celtic Sidhe. Unless you’re dealing with a modern setting (and even if you are) mythology has already developed solutions. It’s a great place to start your search.
However, here are some things I’ll point out:
Cold Iron/Cold Steel are a reference to a specific forging technique rather than a type of metal, though in folklore it can just mean steel swords. Still, this will open up your options some.
Cold Iron for fairies dates back to when iron forging was still mostly new, or less common. There’s certainly lore out there with mythological fairies fighting warriors wielding iron blades, but were unbeatable until new forging techniques were developed.
Ask yourself: is it the forging technique which makes these swords dangerous to your fairies or is it the metal itself? In which case, then you can cut out “cold” as it’s just steel.
Here’s the Wikipedia article about iron in folklore. It may help you some in your search.
If you want to write Urban Fantasy with fairies then I’d go with the forging process rather steel itself. The reason is that they couldn’t go anywhere. At least, not places like the US or Europe or anywhere there’s a high steel content in the buildings, cars, and sewer systems. Even with a shift to polymers too much of the major metropolitan centers in the developed world are built on steel bones. Science fiction fairies re-emerging in the future where all metals are polymers has more potential.
Honestly, any army from a period using steel or iron weapons could curb stomp fairies if they’re allergic to the metal. Using the forging process moves all to some and then down to almost none, making way for the future fairyocalypse of 2018.