That's a ridiculous situation and one which would be annoying and uncomfortable for any other clients in the salon. I would expect the salon owner/manager to take control in that situation if I was another client.
Kids who won't take any notice of their mum often respond to a small ticking off from someone in charge. Could you not say to their mum would you like me to just quieten them down a bit? I bet she'd jump at it!
A kid ran riot in my house once (I do nails from home) and I didn't hesitate to call out for them to come and sit by mummy - how dare they run off upstairs wtf. My kids wouldn't have dreamed of doing that and I was even a bit of a softie but they were taught to have a bit of respect for other people and their things
Sorry for the whinge lol people like that make me cross!!