With funding environment at the minute difficult and priorities for new investment being Crossrail 2, bakerloo line extension etc - this is blue sky (crayon) thinking...
But I’ve always thought the way to increase the utility of this link and to overcome the problems of short platforms, the limited turnaround capacity of Moorgate, the barriers of Crossrail & Bank of England vaults at the southern end was to re route the line, cutting holes into the exiting tunnels as done recently at Kennington for the NLE. I would do this from just north of old street creating new longer accessible platforms at old street, then to divert in a new tunnel to create an interchange at Liverpool st, then onwards in tunnel to London Bridge where it would continue to a tunnel portal south to take over a suitable line southwards. I think you would close Essex road, and incorporate platform lengthening into a much needed rebuild of Highbury and Islington which the only other underground platforms with this constraint. Don’t imagine it would be cheap but it seems to me that like with the LO/East London line you would create a new, high capacity cross London link by joining existing bits of infrastructure. You would create a direct connection between liv st and London Bridge, relieve the northern line between LB/city and improve accessibility of the line. Little map to illustrate as well
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