From the bus ticket kiosk near the post office, Upper Egypt Bus Co has services to Cairo (LE85, five hours) at 6am, 8am, 10am and 3pm. They are often full, so it's strongly advisable to buy tickets the day before travelling. There are two more Cairo-bound buses that originate in Dakhla and pass through Bawiti around noon and midnight, stopping at the Hilal Coffeehouse at the western end of town. Tickets can only be bought on the bus, so hope for a seat.
If you are heading to Farafra (LE30, two hours) and Dakhla (LE50, four to five hours) you can hop on one of the buses headed that way from Cairo. They leave Bahariya around noon from the ticket office.
Travel by microbus between the oases was not allowed for foreigners at the time of writing.