Do you not have a local park, traffic-free, where you could teach him and give SIL a solemn promise that you'll not go near a road?
If she still refuses, at least you'll have eliminated traffic danger as a genuine objection.
All kids deserve to ride a bike: don't give up on him yet
Firstly, has it occurred to you that someone with recurrent chest infections may not be able to do a lot of exercise? For all you know, she may have some chronic underlying health problem that affects weight and affects mobility. Also the fact that she's overweight is *not* as simple as calories in less than calories out - food is one huge emotional minefield and if you display your attitude to her - the one you've shown here, you may well be compounding the problem rather than helping. The only person who can tackle the weight issue is the woman herself, and it may be that right now, she's at a place in her life where she can't. That doesn't mean she's somehow defective or of less worth as a person. Indeed, by backing off about her weight, you could end up doing her a huge favour and she may get to the point where she feels she can tackle her weight problem a shole lot quicker than being nagged about it. I know - I've been there - I was that obese person.
As for teaching her child to cycle: her attitude that it's too dangerous is one that is, unfortunately, held by too many and it's got naff all to do with the weight of the person holding the view. When I allowed my son to cycle to high school each day I got earache from several parents on the 'dangers' and how they'd never let their offspring on the roads as it's 'too dangerous out there'. I would not have been surprised if I'd had a visit from social services due to some of the comments I got from other parents and, in some cases, teaching staff.
The bottom line is that it's not your child so you have no right to unilaterally impose your wants upon another person's child, especially when the parents concerned have said they don't want you to do something. So back off - that way, you *might* get a positive response, but just have to wait longer to get it. As it is, if you nag & push, you are well on the way of alienating a part of your family, and that's not going to get a positive response.
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...
Have you considered asking if you can look after the kid whilst she goes to the gym/swimming/walk/slimming club? The well being of her son is inextricably linked with her own well being and if she feels good about herself and starts to lose weight, she will open up a bit and be a bit more flexible IMO. You then get to spend a bit of quality time with him and can influence him positively. It doesn't necessarily have to be through a bike, but exercise generally would be a good thing (park, walking, swimming, etc). Yes, he's not your son but you seem to care so why not go the whole hog?
Where's Dad in all of this (as has been said)?
As my wife once said, you may enjoy cycling, but that doesn't mean everyone does, nor wants to be Lance Armstrong. And she's right of course.
Credit can be given for knowing *something* about the person, but I highly doubt he'll know *everything* about the person. I seriously doubt he'll know *everything* about her current situation healthwise and everyotherwise. One thing I do know is that weight issues are wrapped up in all sorts of emotional stuff and rarely is it pure laziness with nothing more involved.