The tandem club keep a handy list:
https://tandem-club.org.uk/files/information/trains/index.html
Bear in mind that with Avanti you need two bike reservations for a tandem, and their staff refuse to give you two bike reservations with just one ticket to travel. I'm sure that's fine for most people but I sometimes bring my tandem to the other end of the country, to the person I'm planning on riding it with. I've found in this instance you can usually blag getting two reservations from your local ticket office instead.
Also at least for the local trains I wouldn't necessarily pay attention to what the TOC says anyway. In my experience it's better to use Real Train Times to work out what model of train you'll be getting for your journey, and then look that up on youtube. If it has a big bike space like the train I put my cargo bike on in the photo, you might be OK anyway, especially if it's a quiet train and they don't put many staff on the service.
Obviously carriages that force you to hang your bike are a non-starter.
Hope that all helps.