One way of doing this is to load BP Registration Options. This plugin allows you to edit this email as well as allowing you to moderate all new registrations.
Thanks. I already have this installed. But is it possible to use that plugin without having to manually accept new registrations?
No I don’t think so.
The alternative is to change the text by way of creating your own translation of it using Poedit.
See the following:
Customizing Labels, Messages, and URLs
The poedit program, used for editing and creating the translation files is available for free here:
https://poedit.net/download
You could try Paul’s Welcome Pack plugin. It’s listed as not been updated in 2 years but it should still work.
Someone told me it stopped working a release or two back, so no. 🙂