The text below is taken from the "Notes to read before filing in the application form" for firearms or shotgun certificate, as online on the Thames Valley Police website. https://www.thamesvalley.police.uk/ar/applyregister/fao/tvp/firearms-and-explosives-licensing/application-renewal-firearm-shotgun-certificates/
Take note of sections 6 & 7 in particular, which seems pretty clear.
4.You must disclose any relevant physical or mental health conditions that you have been diagnosed with or treated for in the past as this may affect your ability to safely possess and use a firearm or shotgun. Relevant medical conditions which must be disclosed are listed in note 5.Sections 27 and 28 of the Firearms Act 1968 (as amended) specify that in order to issue a firearm or shotgun certificate the chief officer of police must be satisfied that an applicant can be permitted to possess a gun ‘without danger to the public safety or the peace’. Medical fitness is one of the factors police must consider when assessing a person’s suitability.
FORM 201 – 2017 - 1
5.Relevant medical conditions which must be disclosed are:
•Acute Stress Reaction or an acute reaction to the stress caused by a trauma
•Suicidal thoughts or self harm
•Depression or anxiety
•Mania, bipolar disorder or a psychotic illness
•A personality disorder
•A neurological condition: for example, Multiple Sclerosis, Parkinson’s or Huntington's diseases, or epilepsy
•Alcohol or drug abuse
•Any other mental or physical condition which might affect your safe possession of a firearmor shotgun
If in doubt, consult your GP or contact the police firearms licensing department.
6.If you have disclosed a relevant medical condition the police may ask you to obtain a medical report from your GP/specialist. You are expected to meet the cost if a fee is charged for this. If further information is required the police may request and pay for a further report.
7.Where no relevant medical conditions are disclosed the police will contact your GP asking if they are aware of any relevant medical conditions or have any concerns about the grant of the firearm or shotgun certificate. Depending on the reply, the police may ask you to obtain a medical report from your GP/specialist. You are expected to meet the cost if a fee is charged for this. If further information is required the police may request and pay for a further report.