I believe in terms of employment law you remain employed (if that is the correct term) by your parent service but are placed in the the JHC command structure which is tri-service. The same would apply if you were to be sent to another service for training (e.g. RN Harrier boys doing fast-jet training at Valley) or seconded/attached off to NATO HQ, or an exchange with another country.
As for Odiham not being considered an RAF station, I would assume that if it has "RAF" in the name of the station, it is cnosidered and RAF asset.