I think what you will find as happened is that they don’t have a standard document (there is not one) and hence they are not interested in doing it because it involves work! If you took the certificate to them to have it sworn, they get 5 pounds pocket money for each document and it takes two minutes so they would be keen to do it if you have the document.
Each of you need to have your own Single Status Statutory Declaration confirming that you are free to get married and that there is no impediment.
The Declarations should include your name, address, nationality, religion, passport number and occupation; state whether single, divorced or widowed; that you intend to marry in Cyprus; and be signed and stamped by a UK solicitor, stating -solicitor' on the document. They must be sworn in front of a solicitor.
You have to get the declarations drafted by one solicitors firm or do them yourself and then take them to another firm to have them sworn. They cannot be done at the same firm.
If you are a Scottish national then you should obtain a Certificate of No Impediment from the local registry office in place of the Declaration.
Here is the same thing from one particular website:
If you want me to assist you with drafting the declarations, I can do this for a fixed fee as an additional service £50. All you would need to do would be to fill in the gaps and then take them to a local firm of solicitors to get sworn which is done from a fixed fee of 5 pound each.