Lost Gmail inbox, can't restore with TimeMachine
If it is an IMAP account (don't know if GMAIL accepts POP3) I'm afraid the Time Machine is useless because those mails were not in the local storage - your HDD or SDD, so there is no way Time Machine keep it in backup files. This emails was in a remote Google server and you will have to try their recovery tools.

Usually, the server will keep the deleted messages for a while, even not showing in the trash. This is good place to start: Google Support