I think the info is all out there if one isn't afraid to check, rather than simply assume trains carry bikes.
Hmmm ... for many people, it come from knowing how to check, which is not obvious! [noting Rickwarwicker's follow up - not on the journey planner so you need to find the .pdf timetable online. Simples??]
In theory it should be indicated in the timetable. It isn't always. (Such as the time not too long ago when I tried to take my bike on the service which the timetable said would be a reservation-free Turbo, but a 180 turned up. The guard did eventually let me on, but not without a sternly-delivered lecture.)
I'm noting that an
HST▸ is running one diagram of Taunton to Cardiff today (odd when there should be more 166s about?) and so I expect there will be similar issues there is the train manager is a "Jobsworth".
This issue of cycle reservations was brought home to me at Chippenham a in the autumn. Passenger with cycle waiting to travel on the 17:53 to Trowbridge ... 17:53 cancelled and he had to rush over to the booking office to get a cycle reservation - although it was announced that tickets "via Melksham" would be honoured via Bath Spa, no such concession is offered for people with cycles who plan to travel on units that are cancelled and fall back to HSTs!