In my distant youth if I got hold of an old bike with no logbook, never bought, you notice, mostly free, or 10bob, I would apply to the old LCC for one; They would go through records to find and contact previous owner; if no owner they give you another ,ON THE OLD EXISTING REG number. Sometimes I would receive the old logbook from the previous owner,often with some history.
I have even bought an unregistered scrambler, made an arrangement to ride it, untaxed, to County Hall SE1 where someone would come out to it, cursilory inspect it for frame and engine numbers and a little while later on payment of road tax I'd get a logbook with a reg number, all sorted.
Life is no easier these days, just over complex. Nor safer, nor honest.