I had Botox at 5pm. Can I dye and cut my hair the the next day at 5 pm?

It is fine to proceed to have your hair cut and colored as planned.  There is no concern of migration from the having your hair washed or styled.  Your Botox treatment will be in full effect two weeks from injection.

I would be fine to have your hair cut and dyed the day after a Botox treatment to your forehead.  Washing your hair and massaging your head should not affect your Botox results.