What the installer is supposed to set for the 24-bit code is the Mode S code allocated by the national registration authority.
For example the CAA has allocated a Mode S code for every G-reg aircraft; you can see these in the G-INFO database. The FAA has done likewise.
There is an algorithm for converting these 24-bit codes back into the tail number. It is a while since I saw the details but I recall that the encoding of the tail number into the 24-bit integer is done slightly differently for each country. Some of it is quite obscure; unsuprising since 24 bits doesn't allow a straight encoding of anything up to six alphanumeric characters.
One puzzle is whether ATC will ever have live access to the Mode S derived tail number.
Currently, the ATC doing IFR sectors sees a tail number (e.g. N1234X) next to each secondary return blip but this doesn't come from the Mode S return code; it comes from the returned 4-digit Mode A/C squawk which is used as an index into their current IFR flight plan database which obviously contains the tail numbers. I did a visit to W Drayton recently and this is how it works, even if Mode S is available (which it isn't for the most part and won't be for years).
ATC doing LARS sees even less; I vaguely recall (from another ATC visit) that they don't see the tail number next to the blip and have to refer to the appropriate piece of wood which will have the tail number and the squawk written on it. Also, LARS units will be the last to get Mode S, if ever.
Getting back to who is supposed to set the code, I have it on excellent authority that there are clashes, resulting from multiple planes carrying the same Mode S code. These have been witnessed by people using the ground-based Mode S receivers. It's also trivial to change this code, by going into the transponder maintenance pages, and it will be trivial to set it to something carried by another plane of the same type (by looking it up in the public databases) much as, I gather, people already do in droves when using faked number plates on cars to avoid the London congestion charge. It's going to be awfully hard to nick somebody for busting CAS, purely on the basis of a Mode S code. In the absence of a confession they will need radar tapes as well, in other words exactly like they do at present...
Getting back to the conspiracy theory about enroute charges, the radar coverage isn't available at the low level at which most UK GA flies, say 1500ft and below. There are huge gaps and the radar cover required would be massively expensive. It's certainly true that the UK is covered with radar coverage much of which is not available as a service to GA, or to any other civilian traffic, and it's also true that the coverage of secondary radar (if you don't care about having a primary return as well) is much better than the coverage of the primary radar, but this still leaves huge gaps which would make a mockery of any attempt to run enroute charges on this basis. If we are to get universal charges it would most likely be done on the Canadian model where every GA plane pays something like £50/quarter or whatever.