Now that the internal threading tool is set up and the compound slide is set to the 29.5 degrees we want, it's time to pick up that 8TPI thread. It sounds scary to some people, but it's very easy to do and is a good skill to have.
Configure your change gears, or your quick-change gearbox, to cut an 8 TPI thread. Get the threading dial engaging the leadscrew and working.
Engage the half-nut, and rotate the chuck and workpiece by hand, making sure that you're cutting the proper direction. The tool should advance toward the headstock, and the spindle should rotate in the "Forward" direction.
Now use the cross slide, and run the tool in so that you can get into the bore without touching the existing threads. Disengage the half-nut and rotate the chuck until the "1" mark is lined up on the threading dial-then engage the half-nut. Use the cross slide to bring your threading tool back as close as you can get to engaging the thread, but not actually touching. Use the compound slide to adjust headstock-to-tailstock position, and juggle the two slides until you can rotate the chuck by hand, in the correct direction, and the tool bit follows the existing thread with no crashing. DO NOT disengage the half-nut while you do this, it will only cause problems if you're not very familiar with single-point threading. There are easier methods-but this method works with ANY thread, so it's the one I've described here.
Now that you've got your cutting tool aligned AND timed to the existing thread, make a threading pass. Don't use the cross slide for your feed advance-use the compound slide. That way, each division of the compound slide is almost exactly HALF the actual marked value, effectively doubling the resolution of the dial. This is an 8TPI thread so the carriage slide moves FAST! You may want to turn the lathe by hand if you are not confident in your ability to disengage the half-nut on time.
Take a small cut-just enough to hear the tool scraping-and then retract the tool, disengage the half-nut and run the carriage down the ways, shut the lathe off and do another test fit, by unscrewing the 4-jaw and flipping the assembly over again. See if it threads on all the way up to the register-if it does, and the register is undersized still-perfect. If not, get it to thread up to the register. Remember to use the "1" on the threading dial every time you engage the half-nuts. As long as you do not take the backing plate out of the 4-jaw, you do not disengage the leadscrew with the tumbler or by changing the quick-change box, and you don't remove the threading tool, you will be aligned properly to make another threading pass if you use the "1" mark on the dial.
If the register cut in the backing plate is too big, press in that 1/8x1/8" ring I mentioned earlier, and then you'll bore the register-if it's too small, and won't thread on fully because the register hits, it's time to bore that register until it's a very light drag when you thread on the backing plate. Just like doing the threading, unthread the 4-jaw, flip it, and test-fit that backing plate until it goes on correctly.
You should feel the register bore just slightly increase the amount of force needed to thread the new backing plate fully onto the spindle-it should not require two hands to thread on, but you should feel a little difference right there for that last half-turn or so. That's the ideal fit. If it's too big and you never feel that little drag, it will still work but it will not be repeatable if you unthread the chuck, and it can aggravate chatter problems and removing/intstalling the chuck sometimes. Basically, you're building a tool, build it to the best level of quality and fit you can possibly do. Then, you'll never have to worry about it being inaccurate or wobbly later.
So now, you've got the backing plate fitted to your spindle, and threaded on all the way up, the register bore has that tiny bit of drag when you thread it on, and it fits solidly with no wiggle-wobble. You know the fit, the one that makes you grin and remember why you like turning on the lathe. Now, your job is half done!
Remove the 4-jaw chuck. Give up the safety net, it's time to admit you've done a good job and you're ready to move to the next step. get out your new 3-jaw, and clean the register recess out, making sure there are no chips, and no nicks or gouges.
Set up for OD turning, and dial up your favorite facing and turning tool. Face off the new backing plate, just a skim cut, and turn the OD true. If it already has a register cut into the face, check it against your new chuck-hopefully it's just too tight to go on there, and doesn't fit yet. If not, and the new chuck just slaps up on the backing plate-face off that register. It's not registering anything and isn't helpful.
Now, carefully cut a new register to fit, or trim down the existing register to fit your new chuck-you want it to be a "line" fit. Basically, you want the register diameter of your backing plate to be exactly the same size as the register cut in the back of your new chuck. They will NOT fit together without a firm mallet tap, but it should take no more force than that!
Now, bolt your new chuck up to the backing plate, and let the bolts pull the chuck up onto the register evenly. Use a stamp or a punch and mark the chuck body, and the backing plate, with a punchmark to let you know if you ever have to remove the chuck from the backing plate, how it goes back on.
Congratulations! You've just fitted a new chuck to your lathe, and every time you spin it on and off the spindle, it will run true-with the minimum possible amount of runout.