I've been meaning to investigate this, having noticed it back in the autumn.
On the Thames Valley and North Downs trains worked by 165/166 units, as far as I know you can't make reservations and things continue as before -- see
http://www.nationalrail.co.uk/css/cycleleaflet20151.pdf , which is still linked to from
http://www.nationalrail.co.uk/stations_destinations/cyclists.aspx (it says 'two cycle spaces per train' -- in fact some units have two official bike spaces and others don't, and there's usually no problem with one bike per door vestibule as long as you make sure it's not on the platform side when the train stops).
I haven't taken a bike on an 'inter-city' type Great Western train for a few years; it could be that these trains now require reservations again (as used to be the case if you go back 10 or 15 years), and the presence of a poster at Reading saying the same thing as the website might suggest that it's not just a case of poor writing when the website was relaunched; on the other hand I haven't seen any publicity drawing attention to a change, a friend of mine had no problems taking an unreserved bike Warminster-Bath-Didcot and return in October (which is when I first noticed the change on the website), and if the website is obviously wrong about trains worked by 165s and 166s, then it may be wrong about HSTs and 180s as well.
I haven't taken a bike on local or regional trains in the West Country for many years (but note my friend's Warminster-Bath experience above).
Since writing the above I've looked at the pdf timetable for the Cotswold line at
https://www.gwr.com/your-journey/journey-information/train-times-and-routes. This indicates a few trains on which cycle reservation is compulsory (mostly morning peak arrivals at Paddington and evening peak departures), and also indicates which ones are worked by High Speed Trains (which I think includes 180s as well as HSTs); for these it says 'cycle reservations recomended'. I presume trains not so indicated are worked by 165s or 166s. There are some non-high-speed trains on which seat reservations are available -- I don't know whether you can reserve bike spaces on these.
I don't know what the cut-off for making reservations is -- you might find that if you know what train you're coming back on you can reserve your return journey in the morning.
It seems to me quite possible that nothing has changed and whoever wrote the webpage just didn't know what they were talking about -- certainly I hope that's the case, but I can't guarantee it. If you make the journey you're thinking of, I'd be interested to know how it goes.