Dovecot by default doesn't use any "personal IMAP namespace prefix", which clients often call either "IMAP namespace" or "IMAP prefix". With Courier you probably had this set to "INBOX.", with UW-IMAP you might have set it to "mail/". So, the solution is simply to set this field empty and restart your IMAP client. If this helps, but you don't want to modify the clients' configuration, see Namespaces.
If it didn't help, you might have mail_location setting wrong. If it's unset, Dovecot tries to detect where your mail is stored by looking at ~/Maildir, ~/mail, /var/spool/mail/ and /var/mail/ directories. Depending on what you want, Dovecot might have guessed wrong. See TestInstallation for how to figure out what exactly is the problem.
Missing INBOX (mbox)
See if the mails are stored in ~/mbox file. If ~/mbox file exists, UW-IMAP moves mails there from /var/mail/user. Dovecot supports this with snarf plugin.
Dovecot uses different filenames for list of mailbox subscriptions. You'll need to rename these to ones that Dovecot wants (currently .subscriptions for mbox and subscriptions for Maildir). See Migration for more information.
If it's still not working, check first if the problem is with IMAP client or server configuration. Easiest way to do this is to talk IMAP directly. (include the A, B, C):
telnet imap.example.org 143 A login username password B list "" * C logout
If you see a list of expected mailboxes, the problem is with your IMAP client. If not, set mail_debug=yes and look at the logs.