Think of physiatry as being a field of medicine (physical medicine and rehabilitation) and OMM being a therapy. Manual medicine is practiced in varying amounts by physical therapists and physiatrists, especially sports medicine specialists. One reason that D.O.s are as popular in physiatry as they are is because of our manual medicine training.
Phsycians who practice OMM/NMM exclusively are neuromuscular medicine specialists and sometimes work collaborativelywith orthopedic surgeons, physiatrists, and rheumatologists.
Physiatrists are not physical therapists, but do work closely with PTs. A physiatrist examines the patient, orders imaging if needed, prescribes medications if helpful, and forms a differential and final diagnosis. They then write a prescription for physical therapy based on the patient's problem. This may be "increased range of motion" or "neuromuscular balance training" etc. The PT then takes the prescription and formulates a program based on their examination of the patient and the physicin prescription. Early physiatrists may have done much of this rehabilitation themselves, but that is no longer practical.