This bug is being filed on behalf of ###@###.###.
In Java SE 6 the Swing MenuSelectionManager was made per-AppContext, which according to Chris is a correct fix. However, the class javax.swing.plaf.basic.BasicPopupMenuUI has a static MenuKeyboardHelper, which causes menu accelerators to only work for the first AppContext which creates a menu. The MenuKeyboardHelper needs to be changed to be per-AppContext as well, at which point menu accelerators will work properly again.
This is a high priority bug as it is a regression relative to earlier releases.