Visa's

When u leave the UK you wont pass Imigration so they wont stamp your passport. so unless they share info with the USA the AMericans shouldnt be able to tell from the passport. You can get to Latin America without passing the USA if you have a problem, via madrid with Iberia for instance.

They wont have a clue in the US Embassy. Anyway, if the visa is denied for whatever reason (and there are many), you can fly from Johannesburg to Sao Paulo and from Sao Paulo to Mexico City.

Thanks for the advice, you've been really helpful so far. We are hoping to get there on a round the world ticket as we're looking to backpack through South East Asia, Australia then South America.

We've got enough funds in our account but are concerned about when it asks on the application form about address and place of work. As she has been working in the UK illegally we can't exactly state that on the application form. Her mum owns a shop so we might say she's been working in the shop but we won't be able to prove it if they ask further questions.

She has two blank passports - wonder if they are linked on a system?

#2 when the person leaves the UK they WILL pass immigration. Immigration MAY mark the passport as an overstayer.
Separately, the USA may think that they know everything that is going on in the world, but unless there is an explicit clue and someone happens to notice, so details like duration working won't be a drama.

Be honest. No one in the US cares much about what happened in the UK (if not criminal), but being caught lying IS a big deal here. If you don't say, just be sure you won't get caught.

I don't believe there is such a thing as a transit visa.