I use such program:

import javafx.application.Application;
import javafx.scene.Group;
import javafx.scene.Scene;
import javafx.scene.control.TextArea;
import javafx.stage.Stage;

public class JavaApplication7 extends Application {

    public static void main(String[] args) {
        launch(args);
    }

    @Override
    public void start(Stage stage) throws Exception {
        TextArea ta = new TextArea("one two");
        
        ta.positionCaret(0);
        ta.selectNextWord();
        
        System.out.println(ta.getCaretPosition());
        
        Group g = new Group();
        g.getChildren().addAll(ta);
        Scene scene = new Scene(g, 300, 300);
        stage.setScene(scene);
        stage.show();
    }
}

I see output : 4
and selection till the end of the next word.

I see code of TextArea and TextField behavior classes, and see, that if I press alt+shift+right the selection will be done right. There is condition, whether it is macOs or not which switch selectNextWord() or selectEndOfNextWord(). But method selectNextWord() is open, I can use it, and there is no such condition. I see as a problem solving is to move isMac() condition checking in selectNextWord() function, but not to keep it in keyboard buttons processing, because we have all program analogs of user actions as functions, the same actions, as any user can make and they must work in the same way.