The NHS is not prescribing any particular levothyroxine - the pharmacy is fulfilling the prescription with any of the licensed products.
You can find out much about what is available here:
Regardless of free prescriptions, you cannot legally buy ANY other levothyroxine in the UK without some special arrangements being made. However much you are willing to pay.
It is possibly not a good idea to ask GPs to prescribe a specific packaged form as then, if there is a supply problem, you might find yourself having problems. For example, Almus is simply Actavis packaged in an Almus outer. If your GP wrote Actavis then you could not get Almus and vice versa.
Far better to talk through the issues at a pharmacy and come to an agreement that they will always supply one particular product. If that were to become unavailable, it would be in your hands to discuss what to do with your pharmacist - and does not need you to go back to the GP to get a new prescription.