Yes. Once you click the 'upload to YouTube' button, that video will keep uploading to YouTube in the background and you can do anything else - including recording more videos and exporting data - at the same time.
Once the video is successfully uploaded the link will appear in a grey box. Click on this box to copy that link to your clipboard. You can then paste it say into the Lesson Quiz page in the Laparoscopic Skills Curriculum, eMail it or share it by other routes.
Note: while a video is still uploading a 'progress bar' will be seen while you stay on the History page. If you navigate away from that page, and then return before the video is fully uploaded, the video will still be uploading, but the progress bar will have been replaced by a green box with 'upload to YouTube' in it. This is okay. You do not need to click this again - the video is still uploading. When the video is ready, the link will appear and the box will turn grey.