Yeah, that's a pretty good summary of the problems I have with that design. Great if you've got sawn planks, a brace and bit, and a decent set of chisels (and maybe a fretsaw) - or a chainsaw - but a bit tricky with raw logs, an axe, and a knife...
All I can think is that you could split / hew the planks with the axe (and maybe some wedges), provided your timber is straight-grained and splits well (Sweet Chestnut or Cedar would be ideal). I guess you would have to cut the slot out like you were cutting a mortice. Do-able with a knife, but a lot of work in any timber strong enough to be useful.
Of course, you often see driftwood planks on the coast... That would help for starters. But they tend to be a bit narrow and bendy for this sort of thing.
Come to think of it, timber big enough to make a comfortable chair with this method is likely to be pretty hard to come by in Britain. You're going to need planks at least a foot wide, which means carving up a decent-sized tree. Finding a dead but sound tree of the right sort of wood and the right size might be tricky, and even if you do it's a valuable habitat. And surely none of us would consider felling a live tree for such a purpose...
Maybe a nice thing to make in the shed from bought timber, but I don't see it being field-expedient.