No you don't need planning permission (only need planning permission for outside of your building) but you will need a party wall agreement with all adjoining neighbours. Plus you'll need to get structural engineers drawings and calculations for the steel joist supports you'll most likely need to hold up the remaining stack. You'll also need to get the work inspected by your local councils building control department to ensure the work meets building regs, who will give you a certificate once work has been carried out satisfactorily
I can offer this advice as I had a chimney breast removed in my late Victorian ground floor maisonette.
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