My first reaction is that you don't need to alter the schematic at all. Simply fit the more powerful PSU and change the fan for the hot wire.
What you need to do is check that the MOSFET will handle the voltage from the PSU (which it almost certainly will), handle the maximum current that the PSU can supply, and confirm that it is a true logic-level device. Also, confirm that the Uno will safely handle 15V on Vin.
The hot wire is not inductive, so it shouldn't produce any nasty voltage spikes, which makes your life very much simpler.
I think it will work. HOWEVER, analogRead() will return a value between 0 and 1023, whereas analogWrite() needs a value between 0 and 255. So really if you just take the value from analogRead() and send it straight to analogWrite() you will find only the bottom quarter of the potentiometer* will alter the speed; for the rest of the rotation you'll be sending values higher than 255 to analogWrite(), which will be treated as 255.
So, you need to scale the number from analogRead() to ensure it is always in the range 0 to 255 before sending to analogWrite(). Go to the Arduino reference documents online and read up on the map() function. That will do what you want.
Finally, you might not want to map the range 0-1023 to 0-255, in case that makes the wire too hot. You can limit the maximum power in the hot wire by mapping to a lower number: 0-200, or whatever works.
*(Approximately the bottom quarter, if it's a linear pot.)