Planning consent in Lewisham is generally not required if your garage conversion can be called permitted development (pd). It may be necessary to discharge any restrictive covenants pertaining to your modern property, such as that the garage be maintained as a parking area. By changing the original use of your detached garage, you may be required in Lewisham to apply for formal approval.
Even if pd rights have been revoked, you will still be able to follow a scheme however you would need to put together appropriate sketches and ask for approval from householder planning. That would cost £500, plus any design fees. By completing and submitting an application to householder planning, you would be covered should you desire any major structural changes during the conversion. You may also be required to apply for listed building consents or party wall agreements to complete your conversion.