I don't think the idea of making public transport free (at the point of use) is entirely incompatible with a railway operated by the private sector. The company will need some form of income, but that can come in the form of fares, taxes, or a combination.

Operating a railway is not a very profitable business as it is. The National Rail network is transforming into Great British Railways. It's not complete nationalisation, but the government will have a lot more control. I get the feeling that the private train operating companies are generally welcoming of the changes, especially since it means the state will shoulder the financial risk.