For FTTC (Fibre To The Cabinet) the fibre runs to the cabinet (PCP) and not the customer's property. Those on EO lines, and like you I'm one of 75 EO lines in a cluster in central London (SW9 6xx), aren't connected via a cabinet but directly to the exchange. In our cases our connection is fibre to the exchange and then copper (or possibly aluminium) to our homes. For those connected via an FTTC enabled cabinet fibre runs to an adjacent cabinet and copper or aluminium is only used for the final step.
For the most part BT isn't interested in fixing the problem, there's no BDUK funding available for London to help with the cost and having written to the mayor he seems particularly disinterested in helping and is more focussed on helping small business. Nevertheless I believe BT may be doing something to help those in the Rotherhithe area and installing cabs nearer to the end users which could then have an FTTC cabinet added.
As for FTTP (Fibre To The Premises), which is more like what you describe, there has only been a very limited roll out by BT and whilst it would solve the problem of there being no cab is I feel unlikely in the short to near medium term due to the high costs involved.