Mike,
I assume that you mean the cross feed screw. Reference the Atlas 10 10F parts list available in Downloads. It is not necessary to remove or disassemble the carriage (the apron is a part of the carriage, not vice versa) in order to remove the cross feed screw.
First, crank the cross slide to the rear until the 10F-19 Nut runs off the end of the 10F-36 Screw. Then, remove the 10D-262 Nut and 9-103 Handle. Remove the #3 Woodruff Key. Remove the 3/8"-24 Hex Jam Nut. Loosen the set screw and remove the 10F-62 Dial. Remove the 10F-46 Bearing (it has normal right-hand threads). Note that the drawing shows the flange that contacts the saddle casting as being round. I've seen a photo of one on a 1956 Craftsman (uses the same carriage as the 10F) where this flange is hex. If yours is hex, use a box end wrench instead of a pin spanner. If it is round, you will have to have a pin spanner (and probably a small hammer) to remove the bearing. I don't have a 10F or early 12" and am not certain what the diameter of the flange is nor the diameter of the hole for the pin. But the pin spanner must match both. If two or three firm hammer taps don't dislodge it, spray the joint area with a good penetrating oil and let it sit for a day or so and try again. In either case, once the bearing is removed, pull the screw out toward you, with the 10F-33 Gear on it.
Remove and inspect the gear and key. If OK, transfer them to the new screw. You might want to reverse the gear if it shows any wear on the outer face. I would personally use new woodruff keys. Re-install the screw by reversing the removal procedure. Before re-engaging the screw and nut, adjust the end float by adjusting the jam nut. Check after tightening the outer nut. Repeat until there is minimal drag as you turn the crank.
If you are also changing the 10F-19 Nut, do that after the new screw is installed and before you pull the cross slide toward you to reengage the screw and nut. You will need to remove the 9-37A Guard in order to change the nut. Do not final tighten the 1/4" round head screw attaching the new nut until you have run the new nut fully onto the new screw.