The implementation of IEEE 802.15.6 in Wireless Body Area Network (WBAN) is contention based. Meanwhile, IEEE 802.15.4 MAC provides limited 16 channels in the Superframe structure, making it unfit for N heterogeneous nature of patient's data. Also, the Beacon-enabled Carrier-Sense Multiple Access/Collision-Avoidance (CSMA/CA) scheduling access scheme in WBAN, allocates Contention-free Period (CAP) channels to emergency and non-emergency Biomedical Sensors (BMSs) using contention mechanism, increasing repetition in rounds. This reduces performance of the MAC protocol causing higher data collisions and delay, low data reliability, BMSs packet retransmissions and increased energy consumption. Moreover, it has no traffic differentiation method. This paper proposes a Low-delay Traffic-Aware Medium Access Control (LTA-MAC) protocol to provide sufficient channels with a higher bandwidth, and allocates them individually to non-emergency and emergency data. Also, a Contention Differentiated Adaptive Slot Allocation CSMA-CA (CDASA-CSMA/CA) for scheduling access scheme is proposed to reduce repetition in rounds, and assists in channels allocation to BMSs. Furthermore, an On-demand (OD) slot in the LTA-MAC to resolve the patient's data drops in the CSMA/CA scheme due to exceeding of threshold values in contentions is introduced. Simulation results demonstrate advantages of the proposed schemes over the IEEE 802.15.4 MAC and CSMA/CA scheme in terms of success rate, packet delivery delay, and energy consumption.