I don't know how the QTs handle the events internally, but on most systems at the lowest level the application life goes like this: the main thread code is basically a loop (the message loop), in which, at each iteration, the application calls a function that gives to it a new message; usually that function is blocking, i.e. if there are no messages the function does not return and the application is stopped.
Each time the function returns, the application has a new message to process, that usually has some recipient (the window to which is sent), a meaning (the message code, e.g. the mouse pointer has been moved) and some additional data (e.g. the mouse has been moved to coords 24, 12).
Now, the application has to process the message; the OS or the GUI toolkit usually do this under the hood, so with some black magic the message is dispatched to its recipient and the correct event handler is executed. When the event handler returns, the internal function that called the event handler returns, so does the one that called it and so on, until the control comes back to the main loop, that now will call again the magic message-retrieving function to get another message. This cycle goes on until the application terminates.
Now, I wrote all this to make you understand why sleep is bad in an event driven GUI application: if you notice, while a message is processed no other messages can be processed, since the main thread is busy running your event handler, that, after all, is just a function called by the message loop. So, if you make your event handler sleep, also the message loop will sleep, which means that the application in the meantime won't receive and process any other messages, including the ones that make your window repaint, so your application will look "hang" from the user perspective.
Long story short: don't use sleep unless you have to sleep for very short times (few hundreds milliseconds at most), otherwise the GUI will become unresponsive. You have several options to replace the sleeps: you can use a timer (QTimer), but it may require you to do a lot of bookkeeping between a timer event and the other. A popular alternative is to start a separate worker thread: it would just handle the UDP communication, and, being separate from the main thread, it would not cause any problem sleeping when necessary. Obviously you must take care to protect the data shared between the threads with mutexes and be careful to avoid race conditions and all the other kind of problems that occur with multithreading.