Hopefully the Exeter-Paddington train will remain as it is.
I've taken that train quite a lot with a bicycle, since Exeter is my town of birth!
The train is one of those with a small compartment at the end of the train next to the locomotive, which can take (iirc) six bicycles, theoretically not Tandems, but they do fit (ask Clarion & Butterfly). For this reason, at any time which is remotely likely to be busy, a bike reservation is a very good idea.
At Paddington this is at the end of the train furthest away from the platform entrance, so you need to get there early (the platform number can vary quite a bit). They have been known to put bikes on the locomotive itself, which has a spacious area, but I seem to recall that this is at the discretion of the driver.
Going towards London, it's the back of train, which at Exeter is the left hand side of the station when entering via the main (and as far as I know only) entrance. It's normally Platform 5.
I've never had any problem with this route, but I normally only have a couple of panniers, and make sure that I can easily get them on and off the bike.