Chromatin is the template for the basic processes of replication and transcription, making the maintenance of chromosomal integrity critical for cell viability. To elucidate how dividing cells respond to alterations in chromatin structure, here we analyse the replication programme of primary cells with altered chromatin configuration caused by the genetic ablation of the HMGB1 gene, or three histone H1 genes. We find that loss of chromatin compaction in H1-depleted cells triggers the accumulation of stalled forks and DNA damage as a consequence of transcription-replication conflicts. In contrast, reductions in nucleosome occupancy due to the lack of HMGB1 cause faster fork progression without impacting the initiation landscape or fork stability. Thus, perturbations in chromatin integrity elicit a range of responses in the dynamics of DNA replication and transcription, with different consequences on replicative stress. These findings have broad implications for our understanding of how defects in chromatin structure contribute to genomic instability.