Is she entitled to part of my inheritance once we move in together or get married?

Inheritance rights go to one person at a time. For example, if my father names me his heir and I name my son my heir, as long as I'm alive my son has absolutely no right to my inheritance from my father. So, no one has any rights in your inheritance from your mother as long as you are alive. Second, if you mother has a will, she chooses her heirs, without any limit (except her husband would have some limited rights). She could give her entire estate to the Queen of England, and no one would be able to challenge it (assuming she's not crazy). If she doesn't have a will, her estate goes first to her husband and descendants. Your boyfriend's son is not your mother's descendant