As explained to the hon. Member in the answer given by my hon. Friend the Minister for the Armed Forces (Rt Hon Mark Lancaster) on 13 November 2018, to Question 188918: https://www.parliament.uk/business/publications/written-questions-answers-statements/written-question/Commons/2018-11-06/188918/, citizens of the British Overseas Territory hold one of six types of British nationality which are eligible to apply to join the British Armed Forces, in which they serve as part of the British cohort. There is therefore no requirement to hold discussions with the Home Secretary about the residency requirement for citizens of British Overseas Territories wishing to join the UK Armed Forces.
However, for security reasons the Armed Forces continue to have UK residency criteria to ensure the appropriate security standards which apply to all applicants, regardless of nationality, can be met. This will vary depending on the service and role being applied for, but applications are considered on a case by case basis.