The processing of the I-751 has slowed in tthe last few years. Yours however is the longest I have heard. I also suggest you work with an experienced immigration attorney who can look into processing of the case.
The good news is that your LPR status is retroactive and you will probably be eligible to file for NATZ as soon as you receive your greencard.
Your mistake was to file the I-751 by yourselves, without the assistance of an immigration lawyer to represent you. Am afraid none of this what you are describing would have taken place had an attorney prepared, filed and handled the case on your behalf.
The sad reality is that USCIS does not really care about the unrepresented public. They are only accountable to lawyers and their organization, AILA.
Behar Intl. Counsel 619.234.5962 Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
When waiting for an I-751 interview 12-18 months is not necessarily unreasonable. Unfortunately, these are low priority cases to USCIS. You can certainly retain counsel to represent you and follow-up with USCIS and file a mandamus action, but whether the latter makes sense depends upon the specific facts of your case. I suggest you consult with an experienced immigration attorney who can review your case, advise you as to the options available and recommend a course of action.
While this answer is provided by a Florida Bar Certified Expert in Immigration and Nationality Law, it is for general information purposes only and an attorney/client relationship is neither intended nor created. You should seek out qualified counsel to review your case and provide you with advice specific to your situation. Call +1-561-478-5353 to schedule a consultation with Mr. Devore.
You definitely can do something; you can hire an immigration lawyer. It is not too late to have an experienced advocate on your side who can utilize liaison channels to determine what is happening with your case.