I believe there is a bug in the generated trigger logic. If a NULL column is given a non-NULL value, this won't trigger an update to the tsvector field to include the new text.
The issue is with logic like:
ELSIF (NEW."some_indexed_field" <> OLD."some_indexed_field") THEN do_update = true;
Since NULL <> 'something' is not true in SQL. The simplest solution I could come up with is to replace the <> tests with IS DISTINCT FROM.
Let me know if you'd like a pull request.