Is the situation that she will be working in a salon alongside (but not in a team with) other hairdressers (who are employed and working under a manager), with just her designated workstation to work in (chair, mirror, sink?) and responsible for providing her own clients, tools and materials and determining her own working practices?

She will only be responsible for that which she has control over.

You mentioned law.  She isn't an employer so employer's responsibilities don't apply to her.  The responsibilities of a self-employed person and of a person at work do.  Meanwhile, whoever is responsible for the salon has employer's responsibilities for non-employees under HASAW s.3. 

Who owns the premises is not the critical point in determining responsibilities for its condition; they fall to whoever manages it (eg a hairdressing company).

Your partner should be aware of the risks involved in hairdressing, which are significant enough that the HSE have it as a topic on their website: http://www.hse.gov.uk/hairdressing/index.htm