Our household has been touring with folding bikes since the 80’s. We have used moultons, cresswells, bromptons, dahons, terns and random short lived cheap folding bikes with wheel formats in 16, 20, 24 and 26 inches.
I would concur completely with what has been said up thread about Bromptons. Most of my touring is on Brompton.
I have had a look at Eurostar’s luggage rules and understand them to say that no piece of luggage should exceed 85 cms in any one direction and should be light enough for the owner to carry it without getting in anybodies way. I think that the biggest wheeled bike to easily comply with the 85cm rule would be a 20” wheeled job; there are loads out there on the second hand market. A lot of them are rebadged Dahons, under names like Raleigh, Philips and Dawes. The specifications of 20” folders that I have looked at on line have a longest dimension of about 80cm.
Claude’s first post on this thread also includes the requirements to come in under a second hand type budget and have a low bottom gear; so that looks like buying something cheap and regearing it.
Luggage carriage was also a requirement. Something like a Tern Link or Dahon Mariner would come with rear rack and mudguards already fitted. There are some pannier racks out there which will allow fitting big panniers to a 20” wheeler, but generally smaller panniers will be a better fit. A very useful feature on some folders is a steering tube pre-drilled for the Rixen & Kaul Klick-Fix Headtube Fitting Mount. I have a Carradice Camper Longflap with a mounting block bolted to it which fits the R&K mount.
Gearing can be a bit of a puzzle on folders, due to frame geometries not many can take a front derailler. Schlumpf drives have the problems already mentioned. A folder with a rear OLN dimension of 135mm will take the Sturmey CS-RF3 hub and cassette set up. Most dahons have 130mm dropouts, so it’s a bit of a squeeze. I described fitting one of these to my Tern on viewtopic.php?f=5&t=107139&p=1060751&hilit=tern+node#p1060751
Sheldons gear calculator shows that a 20”x 1.75 406 tyre, CS-RF3 hub , 32/13 cassette and 44t chain wheel gives a bottom gear of 19” and a top gear of 84”.
A point to consider about carrying folding bikes on public transport is that it helps to disguise it a bit. I have sewn up a simple bag from thin nylon (scrap tent material is useful) which just slips over the bike. When challenged I have said something like “it’s sports equipment”, but it’s probably better to be truthful.
Agitate, educate, organise.