Despite restrictions on bikes on trains around the country, it’s still comparatively easy to take a bike on some trains to and from Cambridge. There is no charge for bikes on those direct trains to and from Cambridge which allow bikes (almost nowhere charges any more).
Services to Ely are operated by three different companies, all with different rules. Services to London are operated by two different companies, again with different rules.
Most trains won’t carry specialized equipment (trikes, trailers etc) but all will take folding bikes at any time (though we are hearing reports that some stations – not Cambridge – will not let you through the barrier with the bikle unfolded). Replacement bus services never take (non-folding) bikes (though sometimes you can get sympathetic treatment locally).
The only web site on which you used to be able to make cycle reservations has now disappeared with the demise of the East Coast franchise. You have to visit a station to make a reservation.
London Kings Cross via Stevenage and Royston: Trains from London Kings Cross are operated by Thameslink Great Northern. This is a fast and frequent service. See their current cycle policy. In summary, you may not arrive in London with a bike between 0700 and 0930 (any day) or be on a train that left London between 1600 and 1900 (Monday to Friday). There are also some morning restrictions from Royston to Cambridge, and between Cambridge and Ely. Otherwise you can put bikes in any doorway vestibule, including their trains continuing to/from Ely and Kings Lynn.
London Liverpool Street via Bishops Stortford: Trains to and from London Liverpool Street are operated by Greater Anglia, who also have a detailed cycle policy. Again you can put bikes in doorways, but bikes are not allowed at all on peak services: those arriving in Liverpool Street between 0745 and 0945 or leaving between 1630 and 1830, or to Cambridge from Audley End or Ely between 0745 and 0845.
Norwich via Ely and Thetford, and Ipswich via Newmarket and Bury St Edmunds: Trains to and from Norwich and Ipswich are also operated by Greater Anglia. The rules are different again: trains will take a maximum of four bikes in a designated space (look for the logo by the relevant door – in most of the Norwich trains it is in the middle by the disabled toilet, and at one end on most Ipswich trains) but there are no time restrictions. There is usually little problem getting on these trains but occasionally they substitute trains whiuch will only take two bikes. Note that indirect services to Norwich changing at Ely are operated by Cross Country (see below) which are much more restrictive.
Birmingham, Leicester and Peterborough: From the north and west (via Peterborough and Ely) the operator is Cross Country Trains (see their policy information). These are more restricted: there is only space for two bikes and you are advised to make a reservation. You can book any time up to departure if there is space, which there often is. (Their website implies they will accept bikes without booking if there is space but experience shows you may have problems if you try it).
Stansted Airport: Cross Country Trains also operate direct to and from Stansted Airport. (Note that indirect services changing at Bishops Stortford won’t take bikes at all on the short Stansted Express leg, though it is only a short bike ride between the airport and Bishops Stortford). Bikes can be taken on most planes: many of the low cost airlines operate out of Stansted to cities across Europe and will carry bikes at a modest extra charge. Be prepared to let down tyres and turn handlebars.