You'd need to set up your account as IMAP on both, the phone and Thunderbird. https://support.mozilla.org/en-US/kb/imap-synchronization

Thanks for your reply. I have already set up both Gmail and Thunderbird as imap, which is why I'm confused that the sent folder is not being synchronised with Thunderbird if I send an email using the gmail web interface, or phone.

Please check your gmail account settings.

Tools > Account Settings > Copies & Folders for gmail account or right click on gmail account name and select 'Settings' then select :Copies & Folders for gmail account

On the Copies & Folders tab  : When sending messages, automatically section:

Place a copy in > leave unchecked Gmail auto puts it in the gmail sent folder.

On the Copies & Folders tab in the Drafts and Templates section:

Keep message drafts in: select: Other and [Your Gmail address] > [Gmail] > Drafts

On the Junk Settings tab

Enable adaptive junk mail controls > leave unchecked

Info here; https://support.google.com/mail/answer/78892?hl=en&ref_topic=3397500 https://support.mozilla.org/en-US/kb/thunderbird-and-gmail

Then make sure the gmail sent folder is being subscribed to be visible and synchronised if you need a copy stored in thunderbird.

Huge apologies! I've just realised that the account settings I used was for an email address that was configured to automatically forward to a gmail address and it was the gmail address that I needed to use in Thunderbird! Doh!! It's all working as expected now.

Thanks for everyone's help!