I think it might help if you try to look at it from your father's partner's point of view to see it in context. I guess she thought that if you didn't care for him enough to keep in touch and be part of his (and her) 'support network' while he was alive, why would you bother to travel to a funeral? and indeed, why would she invite you? Grief and relationships are sensitive subjects. We all tend to feel hostility towards people who cause or have caused sadness to the people we love, so if you didn't show much concern for your dad and if this made him sad (because fathers usually do feel sad, deep down, if their sons won't have anything to do with them), it's possible his partner may have resented you for it. But assuming there is no hostility I don't see anything particularly odd in the delay in informing you. It sounds as if your father had built a new life for himself and for whatever reason you weren't part of it. There's a lot to do and cope with when a partner dies, starting with getting in touch with all the people who were part of his life, especially if you were running a business together. Digging out addresses of people who used to know the deceased and need to be informed even though they were no longer in contact, comes later.
So I don't think you should hold a private grudge, and to reply to your question - there is no law that says who must be invited to a cremation, that is a purely private matter and has nothing to do with the legal side of things. The law regarding inheritance is very strict and is not left in the hands of private individuals. There is no reading of the will after the funeral, if that's what you're worried about; the funeral is a private occasion. On the legal level, if you do stand to inherit then the notaire who's dealing with the estate will contact you in due course. Your father's partner has absolutely nothing to do with handling the succession, family members don't take on the role of exector as they do in the UK, they are not involved in it and she has no say over who gets what. It does look very much as if your father arranged things as he did, specifically to ensure that his partner ended up with the house. Legally it is not part of his estate, it belonged to his partner and still does (do you know for a fact that it was even bought with your father's money?). Since you're not related to his partner, you will have no claim on it when she dies, it'll go to her daughter. However, as your father's son you are his protected heir come what may, so I guess you will inherit a share of his non-fixed assets - money in a bank account or savings in his name, his car if it's in his name, etc. Any notaire will tell you the law, but if you wait, the notaire who's dealing with the succession will in due course get round to informing you how the law applies in these particular circumstances. The law is black and white and it will take its course regardless of what you, you father's partner, or anybody else does or doesn't do.