I don't want to muddy the waters but are you sure a contact has to be explicit? When i was looking at employment law for sometime who was working without a contract for a dodgy company, it appeared that a deemed concontr was automatically formed, and if the statement of employment was not provided in writing then the court would decide the terms of the contract.
Obviously that was relating to actually working for an unrelated company, but I'd be surprised if there was no equivalent here.
AsAa different topic this minimum wage thing seems risky t me as well, you'd be lucky to find an electrician happy to work for minimum wage so even then one day the hmrc might start challenging the rates of pay when working for your own company.