Is this a company using land/buildings previously occupied by the school in order to operate a separate business? Have they rented/bought the space they occupy?
Actually, that sounds a but unlikely as you say they have a contract. Who awarded the contract - the landowners? The LEA? Do you know anything else about the terms of the contract? It seems highly unlikely that an individual head has the legal expertise to write a contract without proper legal/procurement support - or that a business would accept one that was not properly drafted/authorised.
Given that the nursery is renting a room from the school I am amazed that you think any kind of open tender process is required.
If I go to a school and say I want to rent a room to do X they will tell me how much it costs. Provided I am willing to pay and they are happy with the use I intend to make of their premises we can do a deal. They do not have to say, "Hang on a few weeks while we advertise and see if anyone else wants to do something similar on the premises".
In this situation it was open to anyone wanting to run a nursery on the school's premises to approach the school and rent the room.
LA rules regarding contracts would only apply if the school was paying the nursery. The fact that the nursery is making (or hopes to make) money from the exercise is immaterial. Any profits they make come from their users, not from the school.