I also am very allergic to nickel. When I first got my wedding ring my finger blistered. I was proposed to on Friday and by Sunday my finger was so blistered that I could not put my ring on. Luckily my jewelers had dealt with this before. They plated the inside of the ring with rhodium so my skin doesn't actually touch the white gold.
New rings made from White gold typically come already plated with rhodium. My ring was my grandmothers that she had made in 1975, so apparently this was not a common practice back then. You should be able to have your jeweler plate your ring for about $50-$100. My jeweler said that you should probably have your ring replated about every 10 years or so because it will wear off.
So luckily you can still wear your wedding ring!