There are two separate issues involved with BOAC operations via European points
(1) is 'cabotage'
(2) is the fact that the German surrender for WWII did not permit them to own or operate aircraft. So all commercial services had to be by foreign carriers.
Under 'cabotage' services to 'empire' destinations could only be served by the national carrier of the non-empire country & the national carrier of the 'empire' country, in other words BOAC. Gradually BOAC set up 'local' national carriers, such as EAAC, WAAC, CAAC, and (via BSAA BWIA) and a host of others.
As these empire countries were granted independence these subsidiaries were handed over to the new governments.
BOAC continued to operate 'via Europe' services on the back of their 'legacy' rights.
The 'Germany' situation continues to allow many carriers to operate 'via Germany' services even today, although mostly no-one bothers, preferring a hub-and-spoke on their own home base. In fact the BA freighters (is it GSS these days?) were still operating via Germany to USA & Canada points using these 'legacy' rights (although now operating, I think, under the "one skies" European Union ticket.)
Rights for all carriers into HKG was for a long time (in theory until handover) under the control of the UK CAA and (notionally) BOAC/BA were the 'national' carrier until the concept of the "national carier" was eroded. (Hence CX into Europe).
I Think about the only remaining 'Cabotage' point is now Bermuda, and in theory BA could set up a hub in Bermuda to serve any points in USA/Canada and link these direct Bermuda-Europe.
Boring aren't I, and probably rongg