I read a story yesterday about this latest Hillary scandal that noted that IMAP software can store copies of all email on any computer that it's used with. I remembered that Thunderbird is IMAP, so I checked. I have every copy of every email I ever sent, or received for work, on FIVE computers at home. The emails date back to 1999. They include confidential emails from Army Tank Automotive Command. This is even after I participated in a technical lawsuit and testified to a judge that I had deleted all emails from that activity. It turns out, they were only deleted from my computer at work.
I am moderately proficient with PC technology, and this came as a complete surprise to me yesterday.
I think that Thunderbird will do POP3 or iMap email. So if you have the setting on your work computer as iMap, yes it will delete the saved emails from all your devices using iMap when you delete from one. Meaning that if that mailbox on your home computer was also set as iMap, then it would delete too. However, if on your home computer Thunderbird was set for this mailbox as a POP3, the saved mail will only delete when you delete it on that specific computer. Hope this is helpful to you.