A data point, I travelled back from Calais to Dover yesterday afternoon by car. No delays anywhere for us. I did see a few coaches queuing, but not many. Don't bother with the duty free, it is the same rip-off as at airports.

That is a bit of an exaggeration. It is usually straightforward on local/regional trains (although there can sometimes be some conditions and also restrictions at peak hours) but, for long-distance/high speed trains and international trains, there are significant variations between countries and it is not always possible to take a bike without dismantling it. On the French network, for instance, only TGVs with designated bike spaces will allow an undismantled bike on board and, on some routes, it is difficult to find any of them at all. Last I looked, there was not a single TGV between Paris and Mediterranean destinations that allowed bikes on (unless dismantled and put in a bag). The cheaper TGVs (Ouigo) will also only allow bikes in a bag, Even when it is allowed, pre-booking is sometimes required.

The key was in the "most" ;)
My point was that it is possible to travel from country to country on trains with a non-disassembled bike, and indeed I've done it between Hungary and Austria, Italy and Switzerland and between Belgium and the Netherlands, a bit like you do today on most UK domestic trains.