Thermal runaway of micro- or nano-protrusions under the high electrical and thermal stresses triggers the electrical breakdown, and which also greatly limits the performances and reliability of linear particle accelerators and micro- and nano-electronic devices. Here, we conduct the multi-physics and multi-scale hybrid dynamics simulations for the mushroom-head like Cu metal nanotips under the radiofrequency and static E-fields to address the possible correlation between thermal runaway mechanism at the atomic scale and the macroscopic electrical properties using ED-MD-PIC method. The simulations reveal that the mushroom-head like Cu nanotip usually experiences a very intense electron field emission stage during the first post half period under a radiofrequency E-field, resulting in a rapid rising of electron and phonon temperatures due to the Joule heating mechanism. The increasing of phonon temperature leads to various atomic structural changes and phase transitions during the heating and cooling cycles, including the bending of nanotip, melting and recrystallization of apex region, and the conversion of the mushroom-head tip geometry to the conical shape. The recrystallization of Cu nanotip from the molten state initiates at dislocation or stacking fault lines at <110>{111} crystallographic directions and planes, leading to the formation of surface <100>, <110> and < 111> facets at the apex under a moderate high E-field amplitude. The electrical pre-breakdown process of mushroom-head like nanotip is characterized by plotting the time to thermal runaway versus the applied E-field amplitude profile, and it is found that a minimum E-field amplitude exists for observing thermal runaway in ED-MD-PIC simulations. This minimum E-field value is about 700 MV/m for radiofrequency case (10 GHz), and which is significantly higher than 525 MV/m for static E-field. In addition, the time to thermal runaway versus E-field amplitude shows a linear relationship.