Many answers from nothing to legal action. I would suggest speaking to your nieghbours and getting them to sign a sort of consent form agreeing to the work. Providing you do not cause any damage to their property all should be OK!! You may also wish to check the support to the nieghbours chimney provided by your half if you remove the chimney completely. If you intend to leave a section of the chimney then it will require support and an application to building control/structural calculations etc..
I agree - make sure you demolish your chimney in the upstairs room or get it supported. And be aware you are going to generate enormous amounts of dust and soot! Remove all furniture and seal up your doors with duct tape before you start.