I talked to the lecturer today. I asked him this question.
He said OH- reacts preferentially to H2O because OH- is a stronger nucleophile. That once all the OH- is used up, bromophenol blue reacts with H+ ions (are they present because of the self-ionization of water?) and a colour change is observed.
Then, my question is, once the OH- ions are used up (their amount is 10% of the molar amount of t-butyl chloride), would water react as the nucleophile, to continue to produce alcohols, as you show in your reaction scheme?
Until all the t-butyl chloride is converted to the alcohol (or the alkene by E1 elimination)