Part 11 of the employment rights act 1996 protects employees from unauthorised deductions from wages. It does not apply where the deduction is made under a provision in the workers contract, where the employee has consented in writing or where the deduction is made as an overpayment of expenses, any of which the employer may claim applies, and an Employment tribunal may decide.
If the “piece of paper” that you describe is an express condition in the contract of employment probably only an Employment Tribunal can decide.
Your wife was aware she has to repay the cost of registration if she leaves within two years of completion and her employers may not decide to take action if she does not. She can stay for two years and then resign, or risk action.
I hope my reply beats that of Builders Mate who may have good advice and sensible comments to make.



