When the screen recorder's teleprompter shows a Document, a new Reload script button fetches the Document again, so edits you have made since opening the recorder appear straight away. The teleprompter also reloads the script by itself when you come back to the recorder's tab.
Reloading keeps your place in the script and the scrolling speed you have set. It is not available while the script is scrolling, so the text never changes under you mid-read.