Is there a term in music to describe what Rex Harrison does in My Fair Lady i.e., speak the lyrics instead of sing them?

It's recitativo. In opera characters sometimes speak to the music rather than sing the melody and that is how this practice is described there.
His effort in My Fair Lady was with full orchestra playing some suggestion of melody. Technically that is called 'recitativo accompagnato'. When it is just done with an instrument or instruments playing 'continuo' in the background it's called 'recitativo secco'.