Recently, the Galactic magnetar SGR J1935+2154 has garnered attention due to its emission of an extremely luminous radio burst, reminiscent of fast radio bursts (FRBs). SGR J1935+2154 is one of the most active magnetars, displaying flaring events nearly every year, including outbursts as well as short and intermediate bursts. Here, we present our results on the properties of the persistent and bursting X-ray emission from SGR J1935+2154 during the initial weeks following its outburst on 2022 October 10. The source was observed with XMM-Newton and NuSTAR (quasi-)simultaneously during two epochs, separated by ∼5 days. The persistent emission spectrum is well described by an absorbed blackbody plus power-law model up to an energy of ∼25 keV. No significant changes were observed in the blackbody temperature (kT BB ∼ 0.4 keV) and emitting radius (R BB ∼ 1.9 km) between the two epochs. However, we observed a slight variation in the power-law parameters. Moreover, we detected X-ray pulsations in all the data sets and derived a spin-period derivative of $\dot{P}=5.52(5)\times {10}^{-11}$ s s‑1. This is 3.8 times larger than the value measured after the first recorded outburst in 2014. Additionally, we performed quasi-simultaneous radio observations using three 25–32 m class radio telescopes for a total of 92.5 hr to search for FRB-like radio bursts and pulsed emission. However, our analysis did not reveal any radio bursts or periodic emission.