Core-shell GaN/AlGaN multiple quantum wells (MQWs) embedded in a p–n junction are integrated on the upper part of GaN microwires grown by silane-assisted metal organic vapor phase epitaxy. Dispersed wires are then contacted by electron beam induced deposition for fabrication of single wire UV-LED devices. Rectifying diode-like behavior is first demonstrated for both GaN and GaN/AlGaN p-n junctions without a MQW active region. A weak leakage current in the GaN/AlGaN core-shell heterostructure is attributed to an additional conduction path along wire sidewalls. Electroluminescence at 340 nm in UV-A is demonstrated using a GaN (2.6 nm)/Al$_{0.3}$ Ga$_{0.7}$N (3 nm) heterostructure embedded in a GaN/Al$_{0.3}$ Ga$_{0.7}$N p–n junction. This value is even decreased to 310 nm by reducing the well thickness to 0.9 nm and increasing the Al-content of barriers (up to 60%) integrated in the GaN/Al$_{0.3}$ Ga$_{0.7}$N p–n junction. This work demonstrates UV-B emission based on single wire core-shell UV-LEDs.