Whilst most of what you’ve said is correct, I actually disagree with you @prh47bridge when you say it would be a straightforward win.
OP, your son could potentially claim for unlawful deductions from wages, failure to provide a s1 statement of particulars (an employment contract), and failing to provide wage slips.
However, the difficulty with cash in hand work is whilst he CAN potentially demonstrate that a contract exists as stated upthread (because it is automatically implied), the problem he could face is that the contract in question could be illegal (cash in hand to avoid tax/NI payments) which means he would lose any statutory right to sue under it. The law requires you to have clean hands, so it will boil down to whether he knowingly/willingly entered into an illegal contract (assuming that it is indeed illegal and that the employer didn’t pay tax/NI if relevant).
If the contract isnt illegal then as @prh47bridge correctly says, I don’t think he will find it difficult to demonstrate that a contract exists.
If he has kept the rotas on WhatsApp, he should be able to demonstrate his hours worked - but without that, he’s effectively asking the employment tribunal to guess what he is owed, which they won’t be willing to do.
He should raise a formal grievance about his unpaid wages in the first instance and follow that procedure, although he might be able to get round that point.
Then there’s the added problem that even if he won, he’d probably never see the money if the company simply doesn’t have it. They will go bust and he will become just another unsecured creditor receiving pennies.
But yes, it’s theoretically possible to sue for lost wages and no, he wouldn’t meet the definition of a modern slave.