The one issue we haven't explicitly covered is that the OP, and any children that are involved as well, need to have their own visas to even travel to the UK if non-tourist entry is contemplated. They can apply for them at the British High Commission in South Africa, but they cannot just all travel together on the wife's visa.

It will be at the visa granting stage that the officials will handle the "can she support you legitimately without recourse to public funds" issue. The prime applicant (wife) will need to provide evidence of funds by showing adequate bank account statements, payslips from the employment that she will use to support them, letters from the wife's employer, everybody's birth certificates, marriage certificate, etc.

Been there, done that.