Optomechanical crystal cavities (OMCCs) allow the interaction between localized optical and mechanical modes through the radiation-pressure force. Driving such cavities with blue-detuned lasers relative to the optical resonance can induce a phonon lasing regime where the OMCC supports self-sustained mechanical oscillations. This dynamic state results in a narrow and stable microwave tone that modulates the laser at integer multiples of the mechanical resonance frequency, ultimately creating an optomechanical (OM) frequency comb suitable for microwave photonics applications. OMCCs enable compact, low-cost power-efficient all-photonic processing of multiple microwave signals, crucial for current 5G and future beyond-5G systems, whilst being compatible with silicon integrated photonic circuits. This work reports the experimental demonstration of all-optical multi-frequency up- and down-conversion of 3GPP 5G new-radio (NR) signals from the low- to mid- and extended-mid bands using the first and second harmonics of the frequency comb generated in a silicon OMCC. The OM comb generates up to 6 harmonics in the K-band, which is suitable for microwave photonic applications. The experimental demonstration also evaluates the impact of the phase-noise and the signal-to-noise ratio (SNR) in the frequency-converted 5G NR signals when the first and second OMCC harmonics are employed for frequency conversion.