I was scratching my head about this concept for ages, but I'm pleased to see now that it works and share it with you all! Here's a way to make Lathe attachments for a drill stand. It's cheap, and you won't have to dismantle an old drill to do it.

One thing I should probably mention before we go on, is that this is essentially a mini lathe. It's certainly not cut out for tough jobs- like turning a quadrilateral form into something circular. This jig is more for adding details, making tool handles from old branches and broom handles etc. Tougher jobs would require tougher bearings, speed control, etc.

Cutting List
Base: 30 (TH) x 30 (W) x 321 (L)

Tools & Parts
22mm Broom Handle
50mm Coach Bolt x 3 (M6)
Pronged Tee nuts (M6)
Wing Nut x 2 (M6)
22mm Forstner Bit
22mm Washer x 2 (M6)
A Sacrificial Swivel Castor (With a matching Socket)
Basic Woodworking tools: Pencil, Try Square, Saw (Cross and Rip), Sandpaper, Centre punch, drill blade set etc.
Uber Basic Metalwork tools: Hacksaw, Rasp set, WD40.

Optional:
Sharpening Jig
Centre Point Marking Jig (See Images 2 & 3)

Step 1: Materials

The first step is to cut and sand your material to an appropriate size. I find that if you use a bench hook you can rest your piece against the edge; which makes it easier to see and remove any discrepancies. The downside of a Verti lathe is that you're restricted to what you can fit between the drill press and your table. A good 6-12" works just fine for me.

Step 2: Marking the Centre Points

I find it helpful to mark the Centre points of your chosen material. This will give your lathe something to grip.

To mark the Centre point on a square, it's as simple as drawing straight lines from Corner to Corner on the end grain (See Image 1). To mark the Centre point of a Circle, it gets a little bit tricky. There's a number of theories in this Great 'iblebut I've found it easier to use a jig.

The Centre point Marking Jig (Image 2)
What you have here is a 22mm hole, to match the width of my chosen material (22mm Broom Handle). The large hole is around 14mm deep (roughly 3/4 of the Jigs thickness), with a 12mm screw inserted from below. If you used a Forstner or Spade bit, there should be a small pilot hole in the centre for you to refer to. You want the screw to fit tightly, and preferably countersunk so you can use the jig without damaging your workbench.
Load your piece into the jig and strike with a mallet. .

Step 3: Swivel Castors

What you want is the chunky pin, which may take a little brute force to extract.Keep hold of its insert too!

Step 4: The Base

This is a platform to support the 'Live Centre' (next step). A mere baton will do

Step 5: 'Live Centre'

Typically, the Live Centre of any Lathe will feature internal bearings; to minimise the friction between the components. My design doesn't have bearings because I felt inclined to use the 'Grip ring' system from Castors. These are cheap, relatively easy to get hold of, and you can turn them till the cows come home and they'll never come out. A little oil here and there will definitely improve their performance though.

Filing these with a drill press is surprisingly easy. My only advice is to keep a steady hand and wear eye protection throughout.

Step 6: The Spur Centre Pt.1

Cutting & Sharpening Jig
I've prepared a jig that'll make this easier (Image 1). No need to write an Instructable for this one, yup, it's simply a 300mm baton with a shorter piece screwed to its end. Take a Coach bolt, insert it into the jig (see picture), secure said bolt with a nut from the other side, then use a hack saw to remove the head.

Once done, you can rotate the jig 90°; which will allow you to file it smooth.

Step 7: The Spur Centre Pt.2

Now that you've removed the head from the Coach bolt, a Pronged Tee nut will be added to grip the work piece. You want to tighten this nut as much as you can, or at least, until it reaches the long smooth area of the shank. To finish, file the other end of the bolt to a point and use an oily rag to remove any residue.

Step 8: Load Up

1) First and foremost, a hole will be needed to take the Spur bit. This will be a breeze, because you should have a notch in both ends of your work piece already. Hold your Spur bit next to it; to see how deep you need to go (Image 1) Remember that the 'prongs/spikes' of the Prong tee nut will be submerged into the wood.
2) Hammer the Spur bit into your piece, then insert it into the drill chuck.
3) Adjust the 'Base' so it's lined up and ready to take the piece from below.
4) To get a tight fit, loosen the head of your drill stand so its entire weight is on your work piece.
5) Tighten up and away you go!

Step 9: Complete

Health & Safety
For those who intend to make this I would strongly recommend taking a good look at your setup beforehand. The milling of metals (which I used to make the spur bit) is purely a cheap, temporary means of achieving a goal; certainly not something to make a habit of. As for the milling of wood, remember people that this is a 'mini lathe', so keep to small pieces of 6-12" softwood. I'd also advise keeping your chisels nice and sharp, to lessen that 'side thrust'. Last but not least, ensure that your drill has an inbuilt fan. If the drill feels hot, let it rest!

Final Note
Feel free to drop a comment if you're unsure of anything! Thanks