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Same here and we have the conveyance from when they were sold off by the local hospital board stating that all shared walls are party walls (and as such shared responsibility). Not sure if that applies to garden walls but it makes sense that it does as they're 1' thick and straddle the dividing line of the properties. Forcing someone to pay their half will involve surveyors and solicitors, it would be easier and probably cheaper to just get it done. If you really want to get them back build it in breeze blocks with some attractive brick slips on your side only (or cladding of your choice).
Ah. House is over 100 years old so not sure it will exist. Where would that be, if it does exist?
Is there any point getting the title plan for next door, to see if there's anything on there?