I’ve wanted to answer this definitively but it’s simply too broad a question! The answer depends on the age of your children, their personalities and needs, teaching styles of the parents, pedagogical method, how present you are able to be, etc.
Generally, when you’re starting out helping children organize, I think you should see how far you can get with simple task lists (either handwritten, printed out or in a text file) until you can discover your needs.
Kanban can work well for older elementary and up, but I don’t know if I’d necessarily use Trello first. They would probably really like to move sticky notes on a wall or whiteboard. Tactility provides benefits to children.
It’s also generally better to avoid burdening them with knowledge of their upcoming work in a central location. So either show them everything they have to do today at once, or show them their future work in a narrow context. For example, put today’s math assignment on the list for today, and if they want to work ahead in math, help them understand how the math book works and what you expect of their worksheets/answer sheets/math essays, so they can just go through the book on their own without being reminded of any other subject while doing that.
When the children are old enough that their coursework begins to be more syllabus-oriented, and/or they start taking external courses, they of course need to start burdening themselves with knowledge of the future. I’d still recommend teaching them to lay out assignments in a paper calendar or maybe a digital calendar before showing them OmniFocus, because they should still be able to work sequentially/soonest due on assignments for the most part.
Just a few initial thoughts that assume some typical needs. It’s an interesting subject to me so I hope to hear more about your plans and challenges.