How to cut a counterbore for chainring bolts with simple tools

First, note that there are three different bolt holes on a 110 mm circle. There's a 5 hole pattern which is symmetric, and there are different Shimano and SRAM 4 bolt hole patterns (...because Standards!)

Fortunately you want a 5 hole, and there's only one common 5 hole format. (But as OJS points out, there's a 5 hole Campagnolo standard that is not symmetrical. It's likely the price of a campy part will make that obvious at purchase time!)

I would look for a countersink bit to fit your drill press, that is the correct side slope, and wide enough to make the hole at the widest end. Ideally it will have a hex shank, not a round shank because they tend to turn in the chuck and lack concentricity.

1. Set a depth stop, by putting the old chainring loosely on a square of flat wood like plywood or MDF or customwood. Its going to get holes in it so nothing special. You may need to pre-drill a pocket or through hole under where the bolt hole ends up, for clearance.
2. With the countersink bit in the chuck but not spinning, lower the quill till it presses on the chainrings hole where you want to bore. Move the chainring till it sits evenly under the bit.
   At this point, set a hard stop on your quill so the drill can't go deeper. If your machine hasn't got a stop mechanism, perhaps try to remember the angle of the handle, or even change the height of the stage so your handle is exactly level at the required depth, or cut a length of wood to stop the handle at exactly the right height. (this is getting to the point where accuracy is going to suffer.)

Now, onto the real chainring.

1. Position the bare chainring under the drill press. Use the same trick as above to locate the chainring under your bit, and clamp down the chainring to the board and clamp that to the stage of your pillar drill. At this point, the handle won't come down as far, and that difference represents the metal to remove.
2. Start drilling. You may need to play with RPM and hand pressure on the handle. Depending on the quality of the countersink bit, you may need to go slow or fast, or with less cutting pressure. Perhaps you may even need some coolant, a slow dribble of water or emulsified oil in water.
   Also the metal of your chainring will affect things. Aluminium will cut faster, steel will heat up which is to be avoided. You are aiming for a smooth-sided hole, not a ribbed one.
3. As you drill, take frequent breaks and try a test-fit of a bolt into the hole. It should fit as flush in the hole as it does into your original chainring.

I suggest you get a couple of cheap used chainrings from your local ebay, or community bike fixup group, and have a go at drilling them for practice.

Other options:

A center finder might help, but they tend to be for milling machines with moveable tables and are overkill here.

You may choose to make a jig, perhaps two pins that just fit through the two adjacent bolt holes. The first lining-up would be no faster, but the remaining 4 should be quicker.

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Some other thoughts:

Keep your existing original chainring as a template. Never ever throw it out.

Another handy resource for chainring shenanigans is the Wolf Tooth BCD PDF. Just print it at 100% no scaling, and check that the 1cm and 1inch test lines are exactly that long. A printout can be stuck to almost anything with spray glue in a can for ease of working.

If you can't find a good chainring candidate with the right 110 BCD hole shape, you may get away with drilling 5 holes around a 4 hole chainring, However you need enough metal to do the job, which might be hard to find.

Finally, if all this sounds too much, there's absolutely no shame in taking the job to a machine shop and explaining what you need. They would probably use a fancy endmill with the same side slope, and run it in a milling machine, perhaps even on a rotary table. It would take longer to explain to them, than it would for them to do the set up. I bet the drilling would be done in a couple minutes tops and won't cost the earth.

What NOT to do - this: