An employee who has received significant training with the Company has resigned from our employment. Can I reclaim the money spent on training during their employment from their final wage?

This all depends on whether or not the employee has been issued with an employment contract that has an express deduction clause in it or has given written consent to the Company to make such deductions.  If there is no such documentation in place, you cannot make the deduction without the employees consent.  Should you make the deduction regardless, the employee will have a claim for unlawful deductions of wages, where they can take you to an Employment Tribunal.