Generally, when a person places an order for something and pays a deposit they enter into a legally enforceable contract with the other side. It is implied that they have accepted the deposit as security and as proof that the customer wants to proceed with the contract.
Unless the provider subsequently commits a serious breach of contract, such as failing to supply the goods or services, or they are the ones cancelling the contract, or there was a cancellation clause, the customer would have no legal right to cancel the agreement. If they do so they will likely be acting in breach of contract and risk losing their deposit. This is reinforced if the contract or initial agreement said that the deposit was non-refundable. Even if this was not explicitly mentioned it is entirely possible for a deposit to be retained if the contract has been cancelled when there was no right to do so.
Whilst there is nothing specific in law which would allow the customer to demand the return of their deposit, it is still worth using some tricks in the book to try and twist the other side’s arm. For example, if initial negotiations have failed, the customer can consider threatening formal legal action if the deposit is not returned. Sometimes exerting some formal pressure like that can have the desired effect. If they still refuse then the only way to challenge this would be by making a claim in court (usually the Small Claims Court) but that could be somewhat risky without the specific legal right to reclaim the deposit.
Another option is to inform them that they would be expected to mitigate their losses by trying to find a replacement customer following your cancellation. So if they are able to get someone else to replace you and the amount you would have paid then that would mean the deposit may no longer be required (or at least part of it) and that the difference should be returned.
The issue now is that you are out of time to take this any further as any claim in the court must have been submitted within 6 years and you have now gone over that, so you will be fully reliant on the company agreeing to refund you, you cannot force them to do so.