if the installer looselayed and nailed the cement board without a thinset base under it, your entire tile installation is incorrect and compromised because he did not give the backerboard a solid base to begin with by electing to looselay it, which is the incorrect method of installing cement backer board.
the correct method(s) of installing cement backer board is in a minimum troweled 1/4" bed of thinset mortar UNDER THE BOARDS and nailed or screwed every 6"-8" and within 1-1/2" from all seams, and some manufacturers allow a bed of high strength construction adhesive UNDER THE BOARDS with the same anchors and naiing pattern, so check with the manufactrer for specifics). This gives the tile system a solid foundation to prevent movement and to flatten or level any minor subfloor inconsistencies that may cause movement in the system that can crack or debond tile and/or grout joints.
Loose laying and screwing is insufficient and does not provide that solid foundation so it would be prone to movement that can cause cracks, tile pops, debonding, and/or cracking grout joints.
The tiles and cement board needs to be gutted down to the bare subfloor, and the cement board anchored properly, or you may have more problems even worse than what you have now.
Basically everything needs to be gutted and reinstalled properly or new product purchased and installed properly if they can not be gutted intact and usable.
If that isnt feasible for you or your workers, then any repairs may be a waste of time, quite temporary, and plan on revisiting this issue time and time again over the years.
Here is some information on CORRECT installation of cement backerboard systems:
How to install Cement Backer Board
In my opinion, your should demand the contractor with 30+ years experience gut all his tile and backerbaord work, install new the backerboard properly, then the rest of the tile work, PLUS foot the bill for anything that is damaged or uninstallable because of damage during the removal process ... because he did a crappy job and must guarantee his shoddy workmanship.
If he refuses, you may have civil recourse becasue his work is improper.
as to removing everything unhamed ... good luck ... i dont think its possible because everything is attached and locked in with thinset.
If you must try delicate intact removale, jsutt go slow and pry lightly with just enough force to bring it up, but again, its improbable becasue you are talking cured thinset, so i doubt the matting will survive.
Q