A measurement-while-drilling (MWD) telemetry system comprises a downhole MWD telemetry tool comprising a mud pulse (MP) telemetry unit and an electromagnetic (EM) telemetry unit. The MWD telemetry tool can be configured to transmit data in an EM-only telemetry mode using only the EM telemetry unit, in an MP-only mode using only the MP telemetry unit, or in a concurrent telemetry mode using both the EM and MP telemetry units concurrently. When transmitting data in the concurrent telemetry mode, the telemetry tool can be configured to transmit in a concurrent confirmation mode wherein the same telemetry data is transmitted by each of the EM and MP telemetry units, or in a concurrent shared mode wherein some of the telemetry data is transmitted by the EM telemetry unit, and the rest of the telemetry data is transmitted by the MP telemetry unit. The telemetry tool can be programmed to change its operating telemetry mode in response to a downlink command sent by an operator at surface.