Losing out on costume jewelry is a bummer, so much kitschy cheapness! But avoiding nickel doesn't necessarily mean that costume jewelry is out the window.
I must have been feeling the red/white when I made this
Plastics and acrylics are always a good choice, and often provide a splash of brightness that metal jewelry might not. There are plenty of non-metal bangles and chunky beaded necklaces to go around! But if chunky/kitschy isn't your style all the time, there's always wood and leather, depending on your personal style.
A lot of mostly-not-metal necklaces still have metal portions or clasps. If you can, attempt a little jewelry surgery! You might try restringing beads on fishing line or ribbon to avoid the problem. If it's charms you're after (like the "Hello!" one in the collage), it would probably be a good idea to splurge on one nice chain of a metal that won't irritate your skin. Usually it's fairly easy to slip a charm off of the chain and switch it onto the nicer one. With one nice chain, you could show off any number of pretty charms as well as create your own unique combos.
I'm not sure of the severity of your allergy, but if you can handle nickel for brief periods of time, you may find a world of accessorizing delight in brooches. They have no prolonged contact with your skin, so should (presumably) be wearable if you don't get a rash putting them on. Plus, brooches are totally fabulous and can land anywhere from classic to obnoxiously rhinestoned on the style-o-meter.
Finally: it usually manages to pop up, but Etsy really is almost always an excellent style resource. There are sellers there who craft jewelry from antique silver, stainless steel (according to the mighty Google, although this is a nickel alloy, it's bound tightly enough that the nickel shouldn't affect the skin), acrylics, wood, leather, cloth, and all sorts of interesting things often for far less money than asked by jewelry companies. I did a quick search for "nickel free" and found lots of adorable and inexpensive things. The sellers there are also usually very communicative and happy to talk about their work, allowing you reassurance that there isn't some secret nickel in those snazzy accessories. There are also many Etsy sellers who deal in supplies, so you can get gorgeous charms and pieces for a fraction of the cost if you're willing to put in a little bit of work on your own.
Good luck, and happy accessorizing!
I don't know a lot about metal allergies, so I apologize if any of this is inaccurate or unhelpful. Do any of you have a tip for our allergic readerfriend?