This is likely to be called permitted development (pd) so the formal planning consent in Southwark would typically not be necessary. Within your house deeds there could be a restrictive covenant which requires that the garage be kept as a parking area, these will need to be discharged. By changing the original use of your detached garage, you may be required in Southwark to apply for formal approval.
By putting together any design sketches and asking for approval from householder planning, you will still be able to convert your garage even if your pd rights have been revoked. Excluding design fees, the cost for that will be £500. If you want to drastically alter the exterior appearance, such as making major changes to the windows, using new materials, or incorporating an extension, a complete application would also be required You may also be required to apply for listed building consents or party wall agreements to complete your conversion.