According to NR Cof C, cycle restrictions are just:
"Subject to individual Train Company charges and restrictions."
So I reckon they are not actually specified, so you cannot claim any generic rights to take a bike?
Where are you getting that from? I'm reading this:
48. Cycles said:Except for a few routes, the Train Companies allow cycles to be conveyed by train. However, restrictions may apply at particular times of day and/or days of the week. Any restrictions will be set out in the notices and publications of each Train Company. A charge may be made for conveying a cycle and a reservation may be required.
Which to me suggests that they can restrict the carriage of cycles by time/day/route but not by ticket type.
In any case, is that actually any requirement for a TOC to always issue tickets that are subject to the NRCoC? Could a TOC not create their own Conditions of Carriage if they so wished?
No as they would be in violation of the terms of their Passenger Licence as granted by the ORR. Such a violation I would imagine would lead to a fine at the very least and continued violation could well lead to them losing their Licence and therefore the franchise.