For the purposes of the Equality Act, any place where a service is provided to the public is a public place. That includes railways and restaurants. A private club might, and I stress the word might, be able to get away with a denial of service or asking a woman who is breast-feeding to leave, but I doubt it. It is against the law for a club, which admits women as members, to refuse them service or discriminate against them on grounds of their sex. Can't see this being treated as an excuse for excluding visiting women who are breast-feeding.