It's worth trying out any sewing thread you can find to see if it works for tatting, it's quite amazing what you can use. Sometimes something will work for chains but not rings (especially if it's a stretchy metallic or polyester thread) and there are lots more possiblities there.
I used to "borrow" the large reels of thread my mother had because she was making cloth dolls known as "Neva" dolls for Hamleys toy shop in London in the early 1970s. Mum could never work out why she never seemed to have any thread!