I was wondering if those "EASA/FAA dual training" providers working in the U.S. holding an ATO approval issued by the CAA UK will be no longer able to train for any EASA certification in case of a "hard Brexit".
There aren't any; all third country ATO approvals are issued directly by EASA. Those schools (such as Naples Air Center) that piggy-back UK ATOs instead of holding their own approval, will not be able to train for EASA qualifications in the event of a 'no-deal' brexit. The same is true of the third country based elements of, for example, CAE and L3.