I think it would be possible in KS2, though it would depend whether you were 'very overweight but capable of normal physical activity' - e.g. walking at a reasonable pace around the school multiple times per day, circulating round the classroom, effectively supervising break duty including going over to a child who was hurt - or 'very overweight and unable to move around the school / class at a normal pace'.
Teaching is a reasonably 'step heavy' job - I wore a pedometer for a day, and just in the course of normal 'collect photocopying, go to another class to talk to the teacher, go down to the front door to meet a parent, walk round the classroom during lessons and setting up, collect resources from elsewhere in the school' activity did thousands of steps per day. It does involve a fair amount of bending and stretching - bending down to pick things up or down to a child, stretching up to get resources stored in various places.
However, it doesn't often require 'long bursts of fast paced exercise', unless your class has a child who 'runs', in which case their individual care plan would need to take into account the fact that you couldn't chase them.
IME most primary teachers do teach at least 1 PE lesson per week - yes, outdoor PE may be taught or supervised by an external coach BUT with 2 hours of PE mandated per week, you will almost certainly be required to teach the other session. That dopesn't require you to be able to DO the lesson, but being able to teach / supervise - move around the room / field, including fairly rapidly to someone working dangerously - would be critical.
Some schools do also have things like the daily mile.
KS1 would be harder, as it is a more 'physical' job and more things will require physical demonstration / will be modelled through physical activity first, as well as more children needing physical help. Again, a minimum of being able to be on your feet all day and be able to bend / kneel / sit down at floor level would be really helpful.
Does that help? Obviously someone with a physical disability would have more 'formal' adaptations made - for example, they might never teach PE or supervise break on the field if it was inaccessible to a wheelchair or walking frame - so adaptations are and can be made. I'm just talking about the main 'what the job entails' points.