Billy you are correct, officers are not saluted if they are not in uniform or wearing headdress. However different rules apply for Royalty, foreign heads of state etc.
Yes, so certain circumstances where a military compliment is required, Prince Harry could be saluted as a member of the Royal Family.
The Army Drill Manual, Section 5 – Compliments, states that military compliments (salutes) derive from the Sovereign and specifically identifies “Members of the Royal Family” as people to whom a form of military compliment is paid.
The fact that Harry is no longer a serving member of the Armed Forces, and no longer holds his former honorary military appointments, does not remove his status as a member of the Royal Family. There is no mention of “working” member sadly.
This courtesy could extend to Zara too, for example, but I don’t think I’ve ever seen a picture of her being saluted.