Why would you need to have wireless control for a "rod wrapping lathe" - it would be much easier with wired control.
You need to identify the motor before you can consider what motor controller would be appropriate. The important consideration is the stall current of the motor.
There are two ways to control the speed normally. Most commercial wrappers us sewing machine motors with foot controller. I have used that type of motor and in fact have two or three sewing machine motors that I can use, but I cannot reverse these without tearing the motor down to switch the brushes. I have also used a 120V light controller to adjust the speed in combination with the foot controller. That works as well, and in fact is a good way to put an upper limit on the motor speed. The problem with this method is, as already said, the motor can't be reversed, and I don't get the really fine speed control I need using the foot controller.
What I want to do is use a DC motor to drive the power head. That gives me the ability to change direction either with switch (DPDT) or programmatically with the Arduino. I have built several rod dryers with dc gear motors and an inexpensive motor controllers, and used Arduinos and the L 298 to do the same thing. But I have not found an easy way to use a foot controller to control the speed. I have built wow wow type foot controllers and connected them to variable pots, but these are not very easy to build and connect, but once done they do provide the control I need.
All that having been said, I have seen several power heads with RF controllers (actually they use the same controller used to control LED lights, similar to what I did the the 120v light dimmer). But these have issues with controlling the speed at the low end. What I want is a controller that give me a little finer speed control at the low end.
The motors I am considering are 12/24 VDC and a max of 2A. I will be powering the motor with a stand alone power supply and a buck regulator to drop the voltage to power the Arduino. All the Arduino will do is provide the RF interface for speed control. I think it will give me the response I need as well as the fine speed control for rod wrapping.
The more I think about it, I think I need to use an IR receiver not an RF module.