It's not that obesity is caused by poverty it's that many of the factors linked to obesity are more prevalent in poorer demographics.
Less food education.
Less access to sporting activities.
Less reward activities that don't involve food (cheap 'treat' in comparison to a new toy or day out at the zoo).
Less stable homes or home life, temp accomedation with unsuitable kitchen facilities or homes without freezers/cookers/money to put in the electric meter to cook lentil soup to feed 100 on a penny.
parents with variable income and hours from zero contract hours, sanctions, poorer money management skills, pay day loans for the birthdays or holidays they don't want their kids to miss out on.
Higher rates of families effected by disability, and the effects range from kids sensory issues like pps have noted to poorer motivation from depression or less organisational skills or being too busy in and out of hospital appointments or caring for dying parents solely, mobility limited or role of medications- they don't 'cause' weight gain in themselves but many increase sedation and carb cravings.
And there is plain dysfunction in terms of parenting skills (choose from lazy neglectful or loving but nieve).
I am not for a second saying all poorer families (I am poor) are like this or that these issues never effect MC families (grew up in one heavily effected by the last on the list) but in terms of general trends these issues are more prevalent in poorer families and they increase risk of obesity, some more some less. There will be others I've missed, sleep education someone mentioned and the role breastfeeding versus formula feeding plays in it also- to do with gut bacteria and breast milk adapting to dietary needs from the little I have read. I suspect these have lesser influence but combined with multipul other factors I don't think it's difficult to see how poverty within a developed country has a role in obesity.
Mini, with regards to your Autistic ds has he been assessed for ADHD? Poorer re-uptake of dopamine is considered to have a role in many neuro developmental disorders but is only medicated for in ADHD. With lower dopamine levels children really have no way of controling impulses and they 'need' that fix in the same way an addict does. ADHD medication addresses this as does one anti depressant that is sometimes used off licence to treat ADHD (only licences for smoking cessation in the UK). When my son went into ritilin his eating stabilised and his sensory sensitivities and cravings became less of an issue. There is a high cross over between ADHD and ASD so you may want to look into assessment. I did read alittle about parents 'naturally' managing dopamine levels with everything from video games to exercise, vitimins and herbal supplements, caffeine and decongestants. I have never tried any of this but you may want to read about it for yourself to make your own decisions.
Someone mentioned fat shaming earlier, how we have a culture of this and parents then end up scared to mentioned weight to over sensitive children because of it. This may be a controversial view but I think fat shaming is mostly aimed at girls. My son did receive some reading when he was 'chubby' but generally it's girls bodies that are scrutinised as they head towards puberty, 'girls' sports where they need to look good in an over sexualised skimpy costume, girls who are fed media images of what their bodies should be and how badly they don't measure up. If society tackled this then we could have more honest discussions about weight as a health issue not as the need to fit into a image that pleases men.