It is difficult to apply for any kind of residency visa if you are on a basic visitor visa waiver.
The point of a visa waiver is you haven't got a visa. You've bypassed the visa system. Therefore you can't change the type of visa you are on, because you havent got one.
Typical you need to leave a country and apply for a visa - you can't get a visa once you are in a country. You can only extend or transfer a visa once you have entered a country on a visa.
If you have no visa to alter you have to go back to square 1 and start the process from scratch.
Immigration never take well to trying to circumnavigate the system. And this is why this lad is so daft.
He easily could have applied for a young persons visa which would have allowed him to work. He chose not to do that. He's not thought about the long term or the practicalities of visas at all.
That's where he will ultimately become unstuck and end up in a right mess - possibly making the continuation of his relationship impossible. And he's done that all by himself. No need for Mum to mess it up for him.
I think the OP shouldn't raise the subject of visas now. Let him work it out for himself as it'll force him to take responsibility.
The danger is he ends up getting himself deported for immigration violations which will affect his ability to enter multiple countries.
He's end up back at home one way or another, because he's not thought it through and he's not going to like reality when it comes and bites him on the bum. Which it will.
OP just needs to be patient because he's not prepared to listen at this point so she'd be wasting her breathe anyway.