Welcome, Simon.
There are three types of brake pads commonly in use today. Fully metallic (often called sintered metallic), semi-metallic, and organic (often called resin). I very seriously doubt that asbestos has been used in bicycle brake pads. If in doubt, you could contact some brake pads manufacturers and pose the question. Shimano, SRAM/Avid, Tektro, and Kool-Stop are four leading brake and brake pad manufacturers.