Mom mixing up clean and dirty dishes... will other family members fall sick?

If she's rinsing out the dishes within a few minutes of being used and rinsing them cleanly (no food bits remaining), probably not much chance of it making others ill from bacteria sourced from the food. If someone is already ill, the chances of passing that illness to others is much greater, particularly cold and flu viruses that can survive on surfaces for several hours.

I recommend placing the clean dishes from the drainer into the cabinet as soon as practical after washing. Would your mother dry them and put them away? (My grandmother always required the dishes to be dried and put away as part of post meal dish washing.) Then when she puts only rinsed dishes in the drainer everyone would know they have not been washed.

Unless someone in your house has a virulent infection and one of you are immune compromised then I'd say the chances of getting sick are between 1% and zero.

Thank you both. Appreciate your help :-)