We present the radio and X-ray monitoring campaign of the 2019/2020 outburst of MAXI J1348-630, a new black hole X-ray binary (XRB) discovered in 2019 January. We observed MAXI J1348-630 for $sim$14 months in the radio band with MeerKAT and the Australia Telescope Compact Array (ATCA), and in the X-rays with MAXI and Swift/XRT. Throughout the outburst we detected and tracked the evolution of the compact and transient jets. Following the main outburst, the system underwent at least 4 hard-state-only re-flares, during which compact jets were again detected. For the major outburst, we observed the rise, quenching, and re-activation of the compact jets, as well as two single-sided discrete ejecta, launched $sim$2 months apart and travelling away from the black hole. These ejecta displayed the highest proper motion ($gtrsim$100 mas day$^{-1}$) ever measured for an accreting black hole binary. From the jet motion, we constrain the ejecta inclination and speed to be $leq$46$^{circ}$ and $geq$0.69 $c$, and the opening angle and transverse expansion speed of the first component to be $leq$6$^{circ}$ and $leq$0.05 $c$. We also infer that the first ejection happened at the hard-to-soft state transition, before a strong radio flare, while the second ejection was launched during a short excursion from the soft to the intermediate state. After traveling with constant speed, the first component underwent a strong deceleration, which was covered with unprecedented detail and suggested that MAXI J1348-630 could be located inside a low-density cavity in the interstellar medium, as already proposed for XTE J1550-564 and H1743-322.