Appoint a Structural Engineer, who will draw up a structural solution for the removal of the chimney breast.
If the chimney breast forms part of a Party Wall ie a wall shared with your adjacent neighbour, then you will need to serve a Party Structures Notice on the freeholder of the adjoining property. Once you are in possession of the structural scheme, you can serve the Party Structures Notice.
If they do not consent to the principle of the works, they have the right to request a Party Wall Award. If this is requested or because the Freeholder has not responded to you with in 14 days of you serving the Notice, you will need to appoint a Party Wall Surveyor to undertake a Schedule of Condition and duly draw up this Award. Once this is in place works may commence. The adjoining owner cannot prevent you undertaking these works.
For peace of mind, consider using one of the RIBA Domestic Building Contracts and check that your building contractor has the required insurance.