BuddyPress.org

You can disable plugins and themes temporarily without losing work. WP and BP content is stored in the database and shouldn’t be affected by what themes and plugins you do or do not have activated.

It’s important that you test whether the problem arises with bp-default, the BuddyPress Default theme that comes with BP. If registration works with that theme but not with your custom theme, then we’ll know that your problem is theme-related. Most likely, your theme is missing the template files that BP expects to find when someone visits the registration page, and for some reason the fallback template (which should be wp-signup.php) is not being found either.

Please test with the default theme and report back.