I don't understand why you think you need to wait for an e-mail. I just did my first export of a gedcom from Wikitree to understand the process and it worked fine with no mention of an e-mail being sent. After you click on Export you will then see a yellow button to Download. Click on that and there will be a bar near the top of the screen giving you the name of your gedcom and an option to save it. Right click on the gedcom name to save it to your computer.
Are you profile manager or on the trusted list for the home person in the gedcom you are trying to download? If not, you will get an error message.