I suspect you're screwed. Because the original design/workmanship has failed inside the expected life of a retaining wall and it is on the boundary the Council will probably insist that they need to be involved in any changes. That will count out anything less than an engineered solution (read new wall properly done). Was the original design approved?

I don't think the position changes the legal boundary just your usable space. I think the wall has to be entirely on the property requiring it. The choice may come down to the cost of excavating the neighbour's side and reinstating.