--that getting a full MRI series of the spine--cervical, thoracic, lumbar--through a private clinic would be very expensive. But given your symptoms, I don't know why the NHS should be putting up such a fight over this (other than being cheap). The doctors there have to know that extremity weakness is nothing to trifle with, and that it can stem from problems anywhere along the spinal cord, as well as from factors that may be affecting peripheral nerves. (It is one of the sad aspects of neurological symptoms that the investigations into them are long and expensive, as very similar symptoms can result from widely divergent causes.)

If standard neurological testing is showing muscle weakness--and by that I mean the simple pushing, twisting, balance movements all neuros should have you perform as part of a normal exam--they really should be delving into this in more detail.