BuddyPress.org

Clearly something is very much not correct is it (the error isn’t just with registration link but all links)? have you taken any steps to rectify the issue that we ought to know about that might help us help you or that we can rule out as potential causes?

There are some pretty standard stock responses given in cases like this to troubleshoot issues that you will see in many posts here.

Fist and most importantly disable BP your site must first work correctly with WP i.e permalinks, when that is established then activate BP but BP only! check if problem is manifest if it is then navigate to the permalinks page doing so flushes caches so check again , still not working? revisit permalinks page change settings and save confirm that you can actually write to the .htaccess file do you see instruction that WP could not save to .htaccess and then list the details that need to go in the file?

if WP and BP work as expected start activating any plugins one by one checking every time to see if problem arises so that if it is a plugin you know which one caused the issue.

I suspect that this is a plugin issue, but those steps above should help you narrow down where the issue is if not correct it.