I thought about it and came up with an idea to solve my problem. Here it is:
"Twist Ties"! Is that what you call them? You know those tiny wires wrapped in plastic that you use to tie your plastic food bags? You can see them at the top of the photo above. Well, they're perfect! Take the plastic off and remove the thin metal wire inside. Fold it in two. Put you thread through it, twist slightly to keep it together and there you have it! Works perfectly for threading seed beads onto your tatting thread.
The day was saved! I could tat away happily. So there you go. If you are stuck without a beading needle, a simple twist tie will do the trick!