I generally either disable email or configure Discourse such that it can’t send email. They will get summary emails if you’re not careful.
You probably want to turn on login required and/or the don’t-index-this-site (I forget the name) system variables so that Google doesn’t index it too.
They’ll need to do a “forgot password” (but you might check out Migrated password hashes support) to get connected with their old account. If someone tries to register for an account with an email address that already exists they won’t be able to since the email address is already a user.
Some number of people will have forgotten what email address they used when they created their account and will whine about it. You’ll have to get them to convince you that they are that person and change the email address for them.