Will this variable voltage supply with variable current limit circuit work?
will it work?
  please feel free to point out any mistakes


The current limit potentiometer just isn't going to work - let's say you want to limit current to 1 amp; this requires a resistance of 1.25 ohms and the power rating of a 150th part of your potentiometer needs to be 1.25 watts. Given that pots have a linear power rating throughout the length of travel, the pot would need to be rated at 150 x 1.25 watts i.e. nearly 190 watts. Sure it will never see 190 watts but right at the end where you might want to limit current to 1 amp, that small fraction of pot track will see 1.25 watts thus the whole pot would need to be rated at 190 watts.

I would also recommend, that from each adjust pin to ground, you put a 10 uF capacitor to reduce the output noise of the rather ancient and noisy LM317 regulators. I'm assuming that you will also need fairly big heatsinks.

If you want a better design try this site called LEARNING ELECTRONICS. On that page they have a design for a fully adjustable power supply that uses an LM317 at the heart of it: -