Squirrel - 809NAS was announced as first RN F-35 Sqn a short while ago. Phoenix from the flames etc, probably to represent the 're-birth' of fixed wing Carrier Aviation for the FAA, whilst sticking with a Sea Harrier & Falklands connection.
The equivalent of the DA and IPA Typhoons are the early jets at Pax River (for B and C variants) and at Edwards (for the A variant). UK involvement at both sites avoids the requirement to have UK specific DA/IPA type aircraft.
One of the benefits of stumping up alot of money upfront to be a level 1 partner....