We investigate high-energy mJ-class diode-pumped acousto-optic (AO) Q-switched alexandrite lasers with broad tunability at both the fundamental near-IR wavelength range and second harmonic generation (SHG) in the UV wavelength range. An AO Q-switched alexandrite laser with continuous-wave diode-pumping has been operated at up to 10 kHz and producing pulse energy of 700 µJ at repetition rate of 1 kHz. With pulsed double-pass diode-pumping, we demonstrate higher pulse energy of 2.6 mJ. With wavelength tuning, this laser system demonstrated broad fundamental tuning range from 719 to 787 nm with diffraction-limited beam quality (M2 = 1.05). By external cavity second harmonic generation in a Type-I LBO crystal, pulse energy of 0.66 mJ is generated at 375 nm and with UV tuning range from 361 to 391 nm. To our knowledge, this is the first demonstration of AO Q-switched alexandrite laser with broad wavelength tuning in the near-IR and UV wavelength ranges and shows its excellent potential as a pulsed source for future applications.