I think this is a hoax.
Chinese state media... not a reputable source, first of all.
Secondly,
HIV/AIDS can only live outside of the human body for at most 15 days in highly-controlled laboratory settings. Fluids lose 90-95% of their viral load when exposed to the air. HPV can also live on outside the human body in the form of fluids deposited on objects (towels, toys, etc), but it’s most commonly transmitted through skin-to-skin contact. The risk of transmission after 15 days, excluding whatever rubber/latex recycling that may exist (and I'm not convinced it does), is tiny. The chances that this is a hoax is high, in my opinion.
There is a chance that these could be recycled UNUSED, expired condoms. There's bound to be a backlog of them hanging around. In general, I have no problem with that idea at all. Recycling is good for you!
EDIT: Who the hell would toss used a condom into a recycling bin after use? And I'm sure we would've already heard about an army of poor municipal workers tasked with sorting said used latex from the rest of the garbage and prepping them for transformation into hairbands? Anyway, latex does not possess the elasticity of rubber, so even if they're unused, expired condoms, you aren't going to get a good product from this. This sounds like an internet rumour, complete with a dodgy "mystery health worker / whistle blower" type.