Imposition is easy as you "only" need to pass legislation for that, *enforcement*, especially for those that do not want to comply... not so much. Mandatory registration at time of purchase - or before you can activate a drone for flight - is certainly doable, if inconvenient for resellers, but that leaves a number of loopholes people could potentially slip through, e.g.:
Hand-built aircraft
Drones purchased prior to the registration requirement
Drones purchased overseas and other "grey" imports
Submission of false details
Secondhand aircraft, depending on how onerous the paperwork is
I don't think anyone with a clue expects legislation and technical measures to completely solve the problem, even if coupled with some stiff penalties for idiots like those at Gatwick today to set an example. However, if they can deter the majority of potential offenders and make it even a little bit easier to identify and prosecute the rest, then that's going to be good enough for government work.