When emerging pathogens encounter new host species for which they are poorly adapted, they must evolve to escape extinction. Pathogens experience selection on traits at multiple scales, including replication rates within host individuals and transmissibility between individuals. We introduce and analyze a stochastic, multi-scale model linking pathogen growth and competition within individuals to transmission between individuals. Our analysis reveals a new factor that quantifies how quickly mutant strains increase in frequency when they initially appear in the infected host population. This cross-scale quantity combines with viral mutation rates, reproductive numbers, and transmission bottleneck width to determine the likelihood of evolutionary emergence, and whether evolution occurs swiftly or gradually within chains of transmission. Wider transmission bottlenecks facilitate emergence of pathogens with short-term infections, but hinder emergence of pathogens exhibiting cross-scale selective conflict and long-term infections. These results provide a framework for a new generation of evidence-based risk assessment of emergence threats.