It would be critical to define or redefine the core values together with the vision and mission which will be embraced by new the employees joining via inorganic acquisitions. The core values should be deployed by the senior management and they have to find cultural ambassadors / leaders to disseminate the culture and vision at every level.
It would be very beneficial to do a co-create session / value mapping together with key stakeholders around the chronic diseases by putting the patient the in the center. Basically, developing a patient journey around the disease and depicting all care cycle would be presenting the inefficiencies and challenges and would help to break up the silos and buy in from all stakeholders.
I fully see and understand the challenge. My suggestion would be creating and developing the platform to pull the team into problem solving environment. This could be a workshop or weekly meeting, where the challenges being discussed with problem solving mode and then assign task / motivate the team to take ownership of the tasks and actions to solve the problem. If you can develop a daily management approach and track the actions weekly and results visa vie outcomes it will create transparency and team work.