If they were welded, there is no way to safely reweld them since I'm pretty sure any welding supplies you'd need to do this easily would be unsafe for birds (like lead sodder)... I think there are safe sodders out there, but they're not easy and would be expensive or hard to find for someone just building bird toys.
Other welding/soddering products might be available like stainless steel, but I'm not sure how easy that is to do, how cheap it might be or the process involved.
You can't just 'blow torch' things back together, its a complicated process involving melted metal and chemicals to make the melted metal stick. My only experiences are making jewelery in high school art classes, and I know it's a tricky, stinky process. Anything smelling like that... I wouldnt let near my birds or let my birds play with.
I have cut a lot of O rings off, and have then been able to use the chain to string new toys on it. I use a bolt cutters my husband has at work if I can't get them with a wire cutter. As a replacement for the O ring I usually use a quick link that is bigger than the hole in the top or bottom toy part (and I make sure the part is indestructable or nearly). That way nothing can slip down off of it and let the toy fall apart.