The lack of paper contract makes no difference - working time is not dependent on a contract and the minimum wage is set by law. I suggest she approaches them directly to start with and advise them that they have failed to pay her the minimum wage for her working time, which is illegal, and that being on probation makes no difference. If they refuse to pay her still, she can consider a free claim in the employment tribunal to get what she is owed.
Before a person can make a claim in the employment tribunal, they would be required to participate in mandatory early conciliation through the Advisory Conciliation and Arbitration Service (ACAS).
The purpose of this process is to allow ACAS to mediate between the claimant and respondent to agree on an out of court settlement in order to avoid the need for legal action in tribunal. The respondent does not have to engage in these discussions, or if they do and the talks are unsuccessful, the claimant will be issued with a certificate allowing them to make a claim.
However, if a settlement is reached, the claimant would agree not to proceed with the claim in return for the agreed financial settlement. Other terms can also be agreed as part of the settlement, such as an agreed reference.
To initiate the conciliation procedure ACAS can be contacted online by filling in the following form (https://ec.acas.org.uk/Submission/SingleClaimantPage), or by phone on 0300(###) ###-####