Nanoscale resistance-switching cells that operate via the electrochemical formation and disruption of metallic filaments that bridge two electrodes are among the most promising devices for post-CMOS electronics. Despite their importance, the mechanisms that govern their remarkable properties are not fully understood, especially for nanoscale devices operating at ultrafast rates, limiting our ability to assess the ultimate performance and scalability of this technology. We present the first atomistic simulations of the operation of conductive bridging cells using reactive molecular dynamics with a charge equilibration method extended to describe electrochemical reactions. The simulations predict the ultrafast switching observed in these devices, with timescales ranging from hundreds of picoseconds to a few nanoseconds for devices consisting of Cu active electrodes and amorphous silica dielectrics and with dimensions corresponding to their scaling limit (cross-sections below 10 nm). We find that single-atom-chain bridges often form during device operation but that they are metastable, with lifetimes below a nanosecond. The formation of stable filaments involves the aggregation of ions into small metallic clusters, followed by a progressive chemical reduction as they become connected to the cathode. Contrary to observations in larger cells, the nanoscale conductive bridges often lack crystalline order. An atomic-level mechanistic understanding of the switching process provides guidelines for materials optimization for such applications and the quantitative predictions over an ensemble of devices provide insight into their ultimate scaling and performance.