Social interaction depends on turn-taking and adapting to one's conversational partner, yet little is known about the typical and atypical development of these abilities. We investigated this in a longitudinal corpus of spontaneous speech in 64 parent-child dyads: 32 typically developing children (20.27 months at start, six girls, 24 White) and 32 with autism (linguistically matched, 32.76 months, four girls, 31 White). Contrary to prior studies, children with autism responded 189 ms faster on average than typically developing children due to more overlapping speech. Latency decreased in both groups (47-78 ms every 4 months) and depended on individual differences in socio-cognitive, linguistic, and motor skills, which for autism explained all variance by age. Both groups equally adapted their tempo to their interlocutors. With robust conceptualization and modeling techniques, we highlight the importance of overlapping speech, show that latencies in autism might be faster than in typical development and situate turn-taking into fine-grained developmental and interpersonal contexts.