You DON'T actually need a JAA instructing qualification - you get an approval issued from the UK CAA.
This provision is for instructors holding a non JAA ICAO licence with an Instructor rating with the relevant experience, who DO NOT HOLD A JAA LICENCE
Appendix 1 to JAR–FCL 1.300
Requirements for a specific authorisation for instructors not holding a JAR–FCL licence to instruct in a TRTO outside JAA Member States or in a FTO partial training outside JAA Member States in accordance with Appendix 1b to JAR-FCL 1.055
To teach PPL and send students solo, you will need an FAA CFI rating as stated.