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I live in London, have a tandem, don't have a car, have done many trips with it on train.
I tend to put it on and ask questions later, the below is based on experience.
Greater Anglia - fine
South Western Railway - fine
Thameslink - fine
Avanti West Coast - fine with a reservation
Great Western Highspeed Paddington - forget it
Great Western Lowspeed Paddington - fine
Scotrail - it depends
Tandems on trains nowadays... Is this a non-starter? Mini c00ps & I were looking at a couple of days tandem packing (nowhere special, just somewhere with some scenery, maybe an outdoorsy activity or two and a quiet countryside route), as I can shift my racks and gear over and we can escape for a couple of days this month or maybe even October half-term.
Last time, a few years ago, I got lucky bringing it back peak time from London to Cambs (either St Neots or Huntingdon - can't remember which station I used back then) on Great Northern, as the gates were open at Kings Cross and I just went through. Not sure about other operators (e.g. changing at Peterborough or Stevenage to go somewhere not on the east coast railway route?
I remember asking similar about a route to Suffolk a few years ago and there was a lot of "no tandems on X service". Assume it's still the same for some trains, regardless of how busy? tl:dr - I need to call the train company(ies) don't I? PITA