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The current behavior of pydoc will cause synopsis to always incorrectly return "None" as the synopsis for any module with mtime == 0. Both of the proposed fixes will fix that bug without affecting any case where mtime != 0, so I don't think either one has backward-compatibility issues.
I'd suggest using the fix of changing the .get call to return a default of (None, None) and changing the conditional to "lastupdate is not None and lastupdate < mtime". That variant seems like more obvious code (since None clearly means "no lastupdate time"), and it avoids special-casing an mtime of 0 and bypassing the synopsis cache.
I don't mind writing a patch if that would help this fix get in. I'll try to write onein the near future, but I certainly won't mind if someone else beats me to it. :)