Outbreaks of tuberculosis (TB) - such as the large isoniazid-resistant outbreak centred on London, UK, which originated in 1995 - provide excellent opportunities to model transmission of this devastating disease. Transmission chains for TB are notoriously difficult to ascertain, but mathematical modelling approaches, combined with whole-genome sequencing data, have strong potential to contribute to transmission analyses. Using such data, we aimed to reconstruct transmission histories for the outbreak using a Bayesian approach, and to use machine-learning techniques with patient-level data to identify the key covariates associated with transmission. By using our transmission reconstruction method that accounts for phylogenetic uncertainty, we are able to identify 21 transmission events with reasonable confidence, 9 of which have zero SNP distance, and a maximum distance of 3. Patient age, alcohol abuse and history of homelessness were found to be the most important predictors of being credible TB transmitters.