I've now spent some time looking at the potential difficulties in this task. Perhaps I should have done this before posing the original question.
I have looked at both the main products of the paper core business, square paper tubes and spiral wound tubes, taking some account of the facilities I have, especially space.
On the face of it square paper tubes are the simplest when small scale maual methods are considered, since a frame using two rollers with nip adjustment and an operator who can acquire the skill to apply an adhesive controllably are all that is needed.
Making spiral cores, except on an ad hoc basis, requires much more thought but, in conversations with ex-colleagues with who have experience can yield consistent results and if needed high production rates.
A problem which the uninitiated often apparently expereince is the diffficulty of removing the wound core from the mandrel. On occasion when making square cores it has been necessary to destroy the core in order to recover the mandrel. This is less of a problem in industrial scale production as the paper tension or the nip pressure on the machine can be controlled as can the condition of the mandrel. The simplicity of a manual technique appeals to me because of its simplicity. Do I have the concentration to make about 50 cores without making mistakes?
As Bill Todd mentioned in his reply I haveexamined the idea of a collapsible mandrel. I am told this should not be necessary (I'm looking forward to proving this). It seems that with the correct adhesive, tension, nip pressure and attention to the condition of the mandrel that a core can be wound which needs minimal force to remove. First picture from the tube gives an idea and in the actual video he manages to remove the mandrel without brute force.
Making a spiral winding machine is much more challenging. I intend to spend some time drawing out how this might be done. Watch this space but don't hold your breath. See the second attached picture from UT for the drive mechanism of this sort of machine. The mandrel in these machines does not rotate but the constant polishing of the paper against the steel keeps it in good slippery condition. Industrial machines need some considerable power to drive the belt rolls, typiccally 40-50 HP depending on the number of plies being combined,