Mammalian cell cultures make a significant contribution to the pharmaceutical industry. They produce many of the biopharmaceuticals obtaining FDA-approval each year. Motivated by quality-by-design principles, various modelling methodologies are frequently trialled to gain insight into these bioprocesses. However, these systems are highly complex and uncertain, involving dynamics at different scales, both in time and space, making them challenging to model in a comprehensive and fully mechanistic manner. This study develops a machine-learning-supported multiscale modelling framework of cell cultures, linking the macroscale bioprocess dynamics to the microscale metabolic flux distribution. As a relevant biopharmaceutical case study, we consider the production of Trastuzumab by Chinese Hamster Ovary (CHO) cells in batch. A macroscale hybrid model is constructed by integrating macro-kinetic and machine-learning approaches. Enzyme-constrained Dynamic Metabolic Flux Analysis (ecDMFA) is adopted to calculate flux distributions based on the dynamic predictions of the hybrid model. Uncertainty estimation of the multiscale model is conducted through bootstrapping. Judging from experimental data, our hybrid model can reduce the modelling error of the macroscale dynamics to 8.0%; a 70% reduction from the purely mechanistic model. In addition, the predicted dynamic flux distribution aligns with observations seen in literature, highlighting important metabolic changes throughout the process. Model uncertainty is maintained at a low level, demonstrating the trustworthiness of the predictions. Overall, our comprehensive modelling framework has the potential to facilitate the development of digital twins in the biopharmaceutical industry.