Hi Dorota,
Sorry for late responding.
I've worked on this and replicated this issue also in my SSAS 2019 environment.
My thought is that in this particular scene, the calculated measure (the sum of two linked measure group) value is processed during the cube processing and stored in the cache (cache of Cube3). When doing writeback in Cube 1 , the linked measure would get its value from linking measure, but this operation could not affect the cached result. So that the m1_m2 measure was not changed.
This is by design , and since linked measure group and writeback are both depreciated feature, I think this would not be reinforced in the future.
Best workaround, my suggestion is, that avoid using linked measure in the cube 3, just physically add direct feature. Since in complicated scenario, linked measure would have a lot of shortages. See: Usage Limitations
If you insist on this structure, the workaround would be like following :
Script a scheduled task in the SQL Server agent, you could set a job to run the cache clear command or cube processing(for Cube3) task as scheduled.
Notice : Clearing the cache to often would affect the cube performance.
Hope this helps a bit.
Regards,
Lukas