I would go ahead and install the coal burner into the existing chimney. Beg, borrow or steal a draft gauge and check the draft you are getting. If this chimney has been working fine with an oilburner, I'm reasonably sure that you're going to have plenty of draft.
What you don't want is a reversing draft. This is when, because of warm outside temperatures, variable winds, and low heat in the chimney flue, the draft is weak, and the combustion gasses and fumes from the coalstove will find it easier to go into the room instead of up the chimney.
If you haven't had smoke and smell issues with the oilburner, I doubt that you will with the coal burner. However install plenty of CO detectors!! Also, don't use other items in the house that use a chimney during warm low draft days. Like another insert in another part of the house. The other items may create a strong draft in their chimney, and pull a slight vacuum on the house if it is a tight house. This will reverse a weak draft in another chimney.
Sort of like the person [ a relative]who didn't check to see that the draft damper was closed on the firplace, then turned on the big house exhaust fan that vents into the attic, which should have windows open to operate correctly, right?? Well the only source of outside air was that fireplace chimney, and she sucked a chimney-full of soot into her living room before she realized what was happening. It wasn't a good day.
Hope this helps, Greg L