The National Rail Enquiries website gives details of the cycle policy for the train company operating the train. For National Express East Anglia it gives the following number for cycle reservations: 0845 600 7245. For FGW the number is 08457 000 125.
http://nationalrail.co.uk/tocs_maps/tocs/LE/details.html
http://nationalrail.co.uk/tocs_maps/tocs/GW/details.html
http://nationalrail.co.uk/tocs_maps/tocs/LE/details.html
http://nationalrail.co.uk/tocs_maps/tocs/GW/details.html