Just had this (or something similar) in Thunderbird 78.8.1 on Ubuntu 20.4.2 after changing a Gmail password - though I'm assuming that "nothing happens" is shorthand for "it doesn't do what I want" : when I clicked "Next" it reverted to the e-mail entry screen but without the autocompleted address, so something happened, but it didn't get me any further.
Steve's workaround ("normal password") gets around things nicely, but I was able to get where I wanted without allowing "less secure" access or uninstalling / reinstalling anything.
It looks like OAuth2 tokens are considered as cookies by Thunderbird, so allowing cookies temporarily in Preferences / Privacy and Security / Web content, deleting both (POP/IMAP and SMTP) stored passwords for that account in Preferences / Privacy and Security / Saved Passwords, coming out of Thunderbird and going back in again will give the same Google prompt with autocompleted e-mail address, but now clicking "Next" prompts for the new password and connects to Gmail, which seems to overwrite the OAuth token.
Cookies can then be disabled again.
[This won't help if you initially had cookies enabled, but I haven't seen the same problem with those settings.]