I have a background in Machine Learning and it is my belief that the mind does use branch prediction. Take for example a scenario where you are climbing down a stair in your house. Your brain already predicts that you are about to land on the next step. Now if the step height was lowered than the usual, the prediction fails and you immediately focus back on the climbing-down process to see what has happened and prosecute whatever fall-back actions are needed to minimize disruption.

An analogy is basically every time you subconsciously predict something, but something else happens. You get surprised and confused which is the brain's analog of flushing the pipeline and calibrating the predictor. Indeed, the execution of most mundane tasks only bubbles up to the conscious level when something unexpected happens and you need to update your model of the world.

Disclaimer: as I said, this is an analogy. Brains are not literally executing instructions in a pipeline.