Your employer doesn’t own you 24/7 outside of work, and cannot dictate what you do in your own time.
Also, it’s pretty normal for most hairdressers/therapists, nail techs to provide their services to their family & friends at home when they’re not working in the salon.
(Although, that isn’t necessarily a good thing as some people will just take advantage and you won’t get a proper day off.)
I don’t see a problem with you doing your mum (or family) assuming you use your own tools and products and you’re not making a profit from it.
If you were actively advertising your services (e.g. via social media) and clearly trying to build up your own business alongside your work in the salon, it would clearly be considered a ‘conflict of interest’ and the employer would be within their rights to ask you to leave.
The tricky situation here is that if you haven’t worked there for long, you won’t have many rights under current employment law and your employer can just let you go.
If neither your mum or you are advertising the fact, I’d say ‘crack on’.