I just pronounce it postgres. That's the original name of the database. It originally had its own query language (quel), and SQL was later retrofitted onto it and called PostgreSQL. But the original quel language is long gone that we may as well go back to calling it just Postgres.
I'm not sure how much that original SEQUEL/SQL has in common with later publicly-available SQL implementations. I never personally worked with SEQUEL but I was under the impression it was more of a spiritual predecessor to SQL than a direct ancestor. But I trust Wikipedia more than I trust my my memory here, so I guess I was wrong.
I actually couldn't find a section on pronunciation in the official FAQ, though I think it was there in the past. Still, they do use phrases such as "an SQLite database", indicating that "SQLite" starts with a vowel sound.