It is important to remember that React expects STABLE keys, meaning you should assign the keys once and every item on your list should receive the same key every time, that way React can optimize around your data changes when it is reconciling the virtual DOM and decides which components need to re-render. So, if you are using UUID you need to do it at the data level, not at the UI level.
Also keep in mind you can use any string you want for the key, so you can often combine several fields into one unique ID, something like ${username}_${timestamp}
can be a fine unique key for a line in a chat, for example.