Being someone who gets on with electricity like water (aka, I break electrics), but is good at building mechanical things, I wondered if I could build my own treadle powered wood lathe for making pens.

What would be the necessary and sufficient conditions to make a small treadle powered lathe that could turn pens and other small (less than 4" across) items?

Is it speed or torque that matters most here?  What kind of rpm would the wood need to turn?  

I would probably use an old sewing machine base, but would I want a secondary flywheel for momentum next to the lathe?

Yes you would. You want to decouple your treadle speed from your spindle speed,as far as I can tell. Otherwise, you're not looking at much of an improvement over the Tudor-era lathe.

But maybe that's what you want. Take a look at the BBC program again. It shows their treadle-powered lathe in detail.

-CK

Those were the branch lathes, weren't they?

Sorry, I don't know the word.  One end attached to a branch, the other attached to the treadle so it goes back and forth.  I'll have to look it up again. Do you have a link?

I was thinking more of circular motion from a treadle and flywheel, then belt driven to a Mandel thingy.  

My dad's lathe is belt driven, but it's run by a motor.  It seems to need a good amount of torque, which is easy to handle by gearing the speed up or down and ensuring there's a good weight on a flywheel to keep the momentum going.  

I think the term you are looking for is pole lathe.  I have a friend that has built one and says it does great.  I haven't seen it much less turned anything on it though.  I've seen old sewing machine treadles turned into lathes as well.  Again, only seen on youtube with no practical experience on my part.  Some episodes of "the woodwrights shop" on pbs has the host using various self powered lathes.  Might be worth a look.

What I'm thinking about is the old singer sewing machines treadle bases have it so the sewing machine head can come out and we can mount something on the same size piece of wood and it fits where the sewing machine head usually goes.  It runs off the leather belt.  This would be easiest for me to use as I wouldn't have to adopt a base or dedicate a base specifically for this tool.  The old 'indian head' spinners are a good example of how the head can be swapped out and the treadle based used for a different tool.

The problem with this that I can see is the belt tension.  The belt is tensioned to the sewing machine and not adjustable.  I think I would want a better belt for a lathe so there would be less slippage (a sewing machine is designed to work with a significant amount of slippage).  There's a kind of belt that is done in links, it's red.  I can't remember what it's called, but it's pretty neat and easy to change the length and is very 'grippy'.  Or I could add something to the system to tension the belt, but that's more moving parts.