Send a message on the booking thread as soon as you know, with the time you now expect to arrive. Ten minutes of warning is the difference between a studio rearranging around you and a studio sitting idle wondering.
Expect one of three outcomes, all of them reasonable: the appointment goes ahead as booked, the service is shortened to fit the time left, or it is treated as a missed appointment. Which one you get depends on how long the service is, how late you are, and who is booked in after you. A studio running fifteen-minute back-to-back appointments has no slack at all.
Being late enough to lose the slot can trigger the studio's late-cancellation or no-show terms, including a deposit being kept. Those terms are the studio's, they are shown on the booking page, and the studio applies them.
If you are late often, book the first appointment of the day or the first after the studio's lunch break. Those are the two slots with room to absorb it.