IMHO, your intentions are correct, and you are headed in the right direction (leaving aside the argument for using QtConcurrency -- thread pools and futures -- since that's not pertinent to the immediate question). Let's address the first part: the objects and execution flow.
As the classes have been outlined in the code snippets, you will need to take extra care to correctly push them across thread boundaries. If you think for a moment, the worker object is created in the calling thread, therefore some of the object's members will also be created in the calling thread. For members which are pointers, this does not pose much of a problem, because you may elect to delay the creation of those objects until after the enclosing object instance has been created and moved to the worker thread. But, embedded objects are created when the object is constructed. If the embedded object derives from QObject, it will have its thread affinity set to the caller thread. In such a case, signals won't work properly. To alleviate this problem, it is often easiest to pass the work thread to the worker object's constructor, so the worker object is able to move all of its embedded objects to the worker thread.
Second, assuming the following:
Command holds a unique instance of LogicHandler, and
LogicHandler does not have state, and
LogicHandler is a subclass of QObject, and
LogicHandler is the worker class
My advice would be to place the spinning up of the thread in Command::redo, then connect the signals similar to advice given at the bottom of this article. Also, you would not set Command.m_outputName to the return value of LogicHandler::run. LogicHandler::run should return void. Instead, you should add a signal to LogicHandler that emits the string value when it has finished processing; then, add a slot in Command to handle that. A QString can easily be marshaled across thread boundaries (make sure you make the connections of the proper type, see here).
The connecting of the worker startup method, to the threads started signal gets the execution started. There is no need to inherit from QThread and override run. The worker should also emit a finished signal, that should be connected to the thread's quit slot. The worker's finished signal should also be connected to both the thread's and worker's deleteLater slot. When these are setup, just call the thread's start method.
From there, the execution will return from redo, and you will be notified that the worker is finished when it emits a signal (the one I mentioned that you will need to add) and passes the output string. If the lifetime of the worker is different (I'm guessing longer, since you need to spin up a thread to do long operation) from the instance of Command, then you will need to connected the return value signal from the worker object to a different object.