It's worth noting that many parts of southern Europe don't differ much from Palestine in terms of Mediterranean climate, and there are native species of the palm family across that region, including Greece, Italy, the Iberian peninsula, and southern France. However, I assume you're referring to northern Europe where palm plants are not native. In that case, they typically didn't procure palm leaves if they weren't grown locally; they used whatever local greenery or vegetation was available. There are plenty of records of other tree branches, such as yew or willow, being used, or various types of flowers. In fact, "Yew Sunday" is not an uncommon colloquial name for Palm Sunday in Medieval English. It seems that there was often plenty of flexibility in the exact material; even at Jerusalem itself, a Christian pilgrim named Egeria in the 380s mentions that their Palm Sunday procession included not only palm branches but also olive tree branches.