Unlike receiving email, many email clients require authentication to send. It's often a different server address. It sounds like the update messed up your authentication token, password, or whatever method they use today.
The update itself probably isn't the issue, just your email settings/authentication.
Assuming you are using a modern email account like Google, iCloud, or Hotmail, you can try removing and re-adding your account in Settings > Mail > Accounts. Don't worry about the pop-up that says it will delete info. That is only on the iPad. It will not go to the email server and delete the online information, When you re-add the account it will sync the information back to the ipad.
Or you can just try signing out/into those accounts. That might work.
If you have an older account, or one from your ISP, you should first make sure it is IMAP or Exchange. The POP3 servers do not keep all your email on the server. At least not normally.