While private health care exists in all of those countries, and in some of them it's a significant part of the system, a lot of what exists isn't actually _for-profit_.
For instance, in Ireland we have a dual public and private system (primary care is private unless you're old, very young, or low income, private hospitals exist as an option), but most of the private system is either outright non-profit (most private hospitals, the statutory VHI health insurer) or is heavily constrained by the existence of the non-profit portion and regulation (private insurers have to compete with the VHI, and are not allowed charge risk-based rates for most customers).