There is no double taxation agreement between Bermuda and the United Kingdom.
The United Kingdom does, however, have a tax information exchange agreement (TIEA) with Bermuda, which entered into force on 10 November 2008.
Details as to what information can be exchanged can be found in the agreement itself. If using the HMRC Intranet, the agreement can be viewed through the “New treaties/protocols in force” link on the sidebar, following the TIEAs link on the subsequent sidebar. On the HMRC web-site, searching for “TIEA Bermuda” will provide a link to the agreement.