On 'normal' public highways, the general speed limits (whatever they are) do not apply to bicycles.
The Highway code references RTRA, which is the Road Traffic Regulation Act 1984, sections 81, 86, 89 & schedule 6. In fact, these sections all fall withinPart VI of the act, which is entirely about speed limits.
The first thing to note is that the Highway Code blanket statement is wrong - section 81 is quite explicit that "It shall not be lawful for a person to drive a motor vehicle on a restricted road at a speed exceeding 30 miles per hour" (my emphasis).
Further reading reveals that throughout this act, the speed limits discussed apply to motor vehicles and their drivers, not to cyclists. Note, for example section 89: "A person who drives a motor vehicle on a road at a speed exceeding a limit imposed by or under any enactment to which this section applies shall be guilty of an offence."
It should be noted that it seems there is a theoretical possibility for a speed limit to be imposed on cyclists. Although the 'standard' way for speed limits is via the RTRA noted above, TSRGD 2002, The Traffic Signs Regulations and General Directions 2002 allows for speed limits to be imposed by local acts (here).
In principle, a local act might create a speed limit that applied to cyclists. However, it's very unlikely, and I've never seen a reference to one that does. Further, the act would have to define the offence and penalty - it could not simply refer back to the RTRA, because the offence created in that act explicitly applies to the drivers of motor vehicles. So, it would be possible for a road to have a speed limit created by local act that applied to cyclists, but it's very unlikely, and even if it did, it is further unlikely that there would be an offence committed in breaking the speed limit.
However, despite the fact that speed limits do not apply to bicycles, you can be prosecuted simply for cycling too fast - under the charge of 'cycling furiously'. You can also be prosecuted for riding dangerously o carelessly. Although very unlikely. Normally you have to kill someone for such a thing to happen. Also since cyclists don't apply to the above regarding the road traffic act a ticket eg a £60 fine cant be issued and as such only prosecution through the court system is possible.