Ultra-wideband (UWB) localization is a recent technology that promises to outperform many indoor localization methods currently available. Yet, non-line-of-sight (NLOS) positioning scenarios can create large biases in the time-difference-of-arrival (TDOA) measurements, and must be addressed with accurate measurement models in order to avoid significant localization errors. In this work, we first develop an efficient, closed-form TDOA error model and analyze its estimation characteristics by calculating the Cramer-Rao lower bound (CRLB). We subsequently detail how an online Expectation Maximization (EM) algorithm is adopted to find an elegant formalism for the maximum likelihood estimate of the model parameters. We perform real experiments on a mobile robot equipped with an UWB emitter, and show that the online estimation algorithm leads to excellent localization performance due to its ability to adapt to the varying NLOS path conditions over time.