I see no mention of the kid's Dad, is he not around? If he was, would he be a better bet to approach?
I can see two sides:
Part of me says walk away. I've got a friend with family trouble and I'm having to tell her to walk away from it, despite her being worried for nieces and nephews, because as far as I can see, she can't do anything and it's just dragging her down too. Although I couldn't concieve of being able to walk away from my nephew, so it's a case of easy to say, harder to do.
Part of me says, get him a bike, teach him anyway. Even if his Mum then doesn't let him out on it, it means he's able to join in mucking about on his mates' bikes, and ride when he's old enough to do what he wants. I'd feel sorry for a lad who couldn't ride a bike, when all his friends could.
I suspect Waffleycat is right though, and the more you push, the more resistance you'll get. It doesn't sound as though the lady is very happy, and unhappy people often become defensive like that because it's easier than changing - I've been there, in my own way. And, espcially if the Dad isn't around, I suspect she may be more controlling because she's afraid that one day the boy won't depend on her and will get a life of his own and abandon her.
Perhaps lay off for a while, then offer a bike for Christmas, then lay off again if refused, and offer for a birthday... At the end of the day, he's not your kid. The best thing you can do is be a positive role model for him (and for the sister-in-law), so that when the time comes for him to be more independent, he sees that there are choices, and that you are willing to help him out with learning etc. After all, it's is never too late to learn a bike...