There seems to be some confusion between obtaining SSR codes for aircraft departing your airfield and CCDS data for code callsign conversion on the radar.
The first requires access to NAS which is difficult because it was never designed to support the type and number of systems that now connect to it. Getting that access is pretty unlikely so you are probably going to be troubling the nearby unit until 2013. After then the UK will be using the European centralised code allocation system and you'll probably be able to get those codes direct from them provided your airport is willing to invest in the systems required to do that. Shortly after that NATS will be replacing NAS with a new FDP system which will support connection to other units as required by the Interoperability Regulations. Again your airport will be able to connect to that if it invests in systems able to do it.
As others have already said NATS does make CCDS data available for code callsign conversion, usually in conjunction with the SSR data but also on its own for those who have their own SSR. There are significant costs in additional equipment, software maintenance and support to allow that access and NATS is allowed to recover those costs plus a contribution towards the system itself to reflect the use of that system. NATS isn't allowed to use en-route charges to subsidise the NSL airports and has to charge them the true cost of providing that service so it seems only reasonable that other airports should pay as well.
The NERL en-route charge does not contain an element to pay for services provided for airports. If you want that to happen you need to lobby the CAA although you might be a bit late for the next charging period.
As part of its contract with NATS for the provision of ATC infrastructure the MOD has already paid its fair contribution towards CCDS and other data feeds.