Hi - My understanding is that the Royalty Fishery – which has references dating back to the 10th century - has been in private ownership since Queen Mary granted the Manor of Christchurch to Francis Hastings, Earl of Huntingdon, in 1554. The Fishery was bought by West Hampshire Water in 1929 and is currently owned by Sembcorp Bournemouth Water. It is presently leased to Southern Fisheries, the agent of which is Davis Fishing Tackle shop in Bargates. (Refer to the Sembcorp website for details.)
I imagine that either Sembcorp or Southern Fisheries would be able to provide more authoritative answers to your queries.
I agree that it is disappointing that public access to this attractive stretch of the Avon riverbank is restricted, but this is obviously a privilege of private ownership.