Nope! The general rule is that, whilst living, fish are considered to be ferae naturae (wild animals) and therefore not in anyone's ownership. The law is a bit complicated and can be subject to interpretation. The carp in your garden pond, for example, cannot swim away and are therefore not deemed ferae naturae. Riparian ownership is, however, much clearer. I may not own the salmon in the river, but if you come come poaching on the stretch over which I have riparian rights or ownership, you'll be up before the beak.
Burnt Ash
The course i did at the Institute of Fishery Management must be wrong then?, mind you it was a couple of years ago, and as you quite rightly say the law is open to interpretation also and i speak from experience, the club i belong to and am on the management committee of, stocked carp, tench and bream into a water that we lease from a local landowner, after a set period of time, and im sorry i cannot remember the period without consulting my fishery management books, the ownership of the fish passed to the landowner despite the fact that it was us that purchased them and could prove this fact by way of a purchase invoice.