We report the generation of mid-infrared (~2 µm) high repetition rate (MHz) sub-100 ns pulses in buried thulium-doped monoclinic double tungstate crystalline waveguide lasers using two-dimensional saturable absorber materials, graphene and MoS2. The waveguide (propagation losses of ~1 dB/cm) was micro-fabricated by means of ultrafast femtosecond laser writing. In the continuous-wave regime, the waveguide laser generated 247 mW at 1849.6 nm with a slope efficiency of 48.7%. The laser operated at the fundamental transverse mode with a linearly polarized output. With graphene as a saturable absorber, the pulse characteristics were 88 ns / 18 nJ (duration / energy) at a repetition rate of 1.39 MHz. Even shorter pulses of 66 ns were achieved with MoS2. Graphene and MoS2 are therefore promising for high repetition rate nanosecond Q-switched infrared waveguide lasers.