I signed a contract to have renovations completed at my property (urgent repairs in relation to our kitchen, bathroom, toilets and laundry). This contract was signed prior to my Khums year as well the deposit for the works paid. The actual work only commenced after the Khums year due to the builder lag time.

Is Khums payable on the amount of money that had been saved to fund the project considering the contract was signed and agreed prior to Khums year?

In this case, since you already enacted the contract with the company and agreed to the amount to be paid, then khums is not due on it.