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Chimney stack responsibility

I seem to recall working on a property with a similar situation, the flue went from one side of the house to the other, it was in or around Croydon. You could try putting some smoke pellets in your fireplace openings & see where it goes.
 
If the flues are serving only them then its their property you cant remove it, unless it somehow falls under the party wall, then it would be shared.
 
The gable end wall with the crack appears to be your gable end as it supports your roof.

You need to take some images in the loft ......it looks to me like your house was originally detached with a chimney stack both ends of the house a usual situation with a detached house. A loft shot illuminated well will provide answers !
 
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