Are there any downsides to receiving BSc, MSc, and PhD all from the same university?
There is an obvious downside of staying at the same university—you will only work with one set of advisors and contacts throughout your career. That will mean that you may not benefit from the experience and diversity of viewpoints and philosophies that comes from working with different groups of people across the length of your career.

However, if there is a compelling reason to remain at your school, that's a different situation. This could be financial, personal, or educational. For instance, a family member may have a job that makes it difficult to relocate, or a new project may be starting up that provides a unique educational opportunity.

Beyond that, though, I would lean towards going somewhere else for your PhD program!