They presumably want to know whether the tenant has permission to run a business from home? You need the correct insurance as well as permission from the council - do they have all of that, and if not, why not?
As far as I can see from googling, the only permission they are likely to need is from their landlord.
I think the council only comes into it for particular businesses that need a license, for example child-minders, not for businesses generally.
Similarly, council might come into it if alterations to the property were required, but presumable that's not the case here.
They may need various types of insurance, but if they don't have relevant insurances, I think that's their problem? I don't think it's generally illegal not to be insured, just unwise.
As the landlord, your mum could get into trouble if her tenants are reported for illegally running a business from a domestic dwelling.
It's not generally illegal to run a business from a domestic dwelling, so no-one is going to be in trouble.