Um, chances are it doesn't matter how friendly Rickman is, he will see it as interference in his relationship.
"None of your business" is the standard kneejerk response to women asking about the new girlfriend, with the implication that the only reason they are asking is because they are sad, nosy, embittered women who haven't "moved on", not because they have any right whatsoever to be concerned about the stranger who may be destined to have quite a strong influence over their children's lives.
If this woman is so committed to Rickman's children (and I can't say I'd be too enthusiastic about another woman making a commitment to mine, thanks very much), then Rickman has a perfect right to at least know a few basic things about her.
And tbh even with a man, I think it's reasonable to ask the question, "what are the care and control and contact arrangements you have with your children and why?" even though it is not as bog-standard for them to have care and control. Any reasonable person who is going to be in charge of someone else's children, wouldn't really object to answering those questions imo, depending of course on the tone in which they are phrased.