First, a bit of history. Cars were first registered in the UK as a result of the Motor Car Act 1903, which introduced compulsory registration of cars, licensing of drivers (although there was no driving test) and a new offence of 'reckless driving'. It also raised the speed limit. The provisions came about partly because of the need to identify cars involved in accidents and crime, and because of the death, injury and intimidation motors were inflicting even back then.
Bear in mind that the bicycle predated the motor car by some years, yet registration of cycles had not been introduced.
Registration for cars makes sense because a car has exception powers of escape. A driver can cause a serious or fatal crash and still be able to leave the scene, it being impossible for anyone on foot to stop the vehicle. The driver is also somewhat hidden from view, particularly if the vehicle is being driven fast. Also, there is the need for the vehicle to be kept in good mechanical condition, hence the MOT test which needs to be correlated with the vehicle's identity. Lastly, registration helps against vehicle theft.
How would registration of cycles work? If it worked in the same way as car registration, where a registration number is displayed on a number plate, it would become easy to identify the cycle and the rider, in the event of an offence (either a road traffic offence or cycle theft) being committed. However, there are some clear practical difficulties. There is no obvious way of displaying a registration plate on a bicycle. For a plate to be readable at any distance, it would need to be a considerable size, and would need to be firmly affixed to the cycle. The obvious place to put the plate would be on the rear rack, but many bikes don't have one. Under the saddle would be possible, but it would likely be obscured by clothing. It could be on the frame perhaps, like the advertising panel on an old butcher's bike, but that would interfere with water bottles, D-locks and the like. The number plate could be in the form of a sticker affixed to the frame, but it would likely be too small to be readable at any distance, and in any case could be easily obscured by the riders leg.
To counter theft, there are voluntary registration schemes already in operation, but they don't rely on a large plate.
How would registration of cycles be paid for? To have a centrally administered system, with registration plates or marks that are difficult to falsify would be expensive, and the costs of registration could easily exceed the price of a cheap bike. Cycles are relied on as transport by some of the least well paid workers, and it would be unfair to expect them to pay for registration retrospectively. It seems likely that the cost of setting up a registration system would fall at least partly on the taxpayer.
What would registration of cycles achieve? In theory it would make cycle theft more difficult, because of the need to change the cycle's identity. There are people who want cycles to be more identifiable in the event of a traffic offence being committed. However, having a registration plate on a car isn't particularly useful unless there has been a serious crash. Try reporting a car jumping a red light, speeding, mobile phone use or even dangerous driving and the police won't prosecute the offence unless there's been a collision; prosecution for similar offences is hardly more likely for cyclists, so registration won't make any difference here, except possibly as a deterrent. Registration doesn't seem to deter many car drivers from breaking the law. In the event of a crash caused by a cyclist, the cyclist is likely to be injured, and in any case it is easy for a member of the public to stop the rider leaving the scene.
What about other crimes committed by criminals on bikes? It's true that the cycle can be used as a means of escape, but it's not much more use than simply 'legging it': a fit and alert member of the public could stop a cycle from leaving the scene of a crime. As a means of attack a bike's pretty useless, as likely to injure the rider as anything else.
There are very few deaths or serious injuries caused by cyclists. In an average year, one or two people are killed by cyclists, and that includes all types of incident including those where the cyclist was not at fault. There's no particular reason to think that this number would be significantly reduced by registration. However, there is good reason to think that registration would act as a deterrent to cycling. All over Britain, there are millions of rarely-used cycles in peoples' garages. Will people be prepared to pay to register them? If not, then they won't be used at all. So the owners won't be able to try cycling to work, or try cycling to the shops, or go out cycling with their children or grandchildren. There will be a clear barrier to people getting into the habit of cycling, and thus fewer cyclists. Given the health benefits of cycling, it therefore seems likely that more people will die or be disabled by the diseases of a sedentary lifestyle than could ever be saved by a registration scheme.
Cycle registration is a concept that is tossed around by people who don't like cyclists, don't respect other peoples' choices and in the main aren't terribly clever. They haven't thought through the benefits, drawbacks, costs or practicalities, because if they had, it would have rapidly become apparent that it's a pretty stupid idea (unless you want to suppress cycling, in which case it's a very good idea). Unfortunately, such people seem to hold an inordinate amount of influence in this country...