Of course charities are audited just like companies.
They are registered with the charity commission which has very strict reporting requirements and accounts and disclosures are required to be prepared in accordance with the Charities SORP, which in my experience, is at least as onerous as companies accounts disclosure requirements.
And in any case, auditing has nothing do with how much the staff are paid. For both charities and companies this has to be disclosed, but the audit has no role in setting any of the rates.