Is there yichud for a man with his niece or a woman with her nephew? At what ages is it permitted? And is the heter under any special conditions?
A woman wish her nephew is a full case of a Torah “erva” (an aunt), and the prohibition of yichud applies. The prohibition also applies for a man with his niece, even if she is not married (yichud penuya), and the more so if she is married.
There is no “special heter” for such circumstances, and the regular principles of yichud (shomrim, keeping the door open, and so on) will apply.
For the principles concerning yichud of children, please see here.