Landlord is refusing to install locks in a shared house is this illegal?

You are living in a shared house, you have entered into a contract with the other tenants and you should trust them not to steal your stuff. Otherwise they could just leg it and leave you on the hook for the rent, or for damages if they break stuff. If the landlord installs locks on bedroom doors that would make it an HMO and that is a fundamental change to the operation of the property. Among other things it needs planning permission, licencing (even for small HMOs in some boroughs), different tenancy agreements, internal building works like emergency lighting, firewalls and fire doors, regular inspections of "common areas", and other changes like no notice period for them to enter those common areas. You can install your own lock or have one installed, but remember you are responsible for returning that to the original state at the end of your tenancy.