In est, you can think of the GG’s appointment as being done by the Sovereign — just with the nomination process delegated to the country’s current PM in order to minimize the Sovereign’s workload (especially for the Commonwealth, where they don’t live anywhere near the country themselves, and so don’t have an independently-legible candidate pool to choose a GG from. Back when Canada first became independent, appointing a GG independent of the PM would have meant a transatlantic voyage!)

Likely, if these countries were only just de-colonizing and ratifying constitutions today — where the existence of an intelligence service with global reach, directly reporting to the Sovereign, can be assumed — then the GG appointment for each country would be probably be done directly by the Sovereign, with MI6 analysts doing the nominations and then the countries’ PMs (or trusted delegate cabinet ministers) merely doing recorded interviews for the Sovereign to watch and evaluate.