How it generally works now is that the operators win a contract from the government to run a line for X years. They pay the government for that contract, then have (as a for profit company) to make enough money to make a profit on top of that. There are various rules though on the fare increases allowed (hah!) and the contracts generally include minimum service levels etc.

The actual tracks and signals are run by network rail who are a state owned company that own most of the infrastructure (replaced Railtrack who were an actual public company that went bust!)

The stations are managed as part of the operator contracts I believe? Unsure on their ownership.

Because the operator's contracts are time limited the operators can change over time, when that happens though the rolling stock (trains) are generally transferred with that, either through being sold to the new operator or leased. (It's not like you can order a new fleet of trains and get them delivered on short notice!)

My understanding is that labour is mostly going to take over the running of each contract as it runs out instead of buying them out. I assume they are then leasing the trains to avoid the up front expense of buying them all. As they won't be trying to make a profit it (should) be possible to run a cheaper service and could/should allow the lines to be run for the public good instead. (That's where people opinion of public ownership comes in)

There are already some lines run like this as the operator went bust and it defaulted back to government control.