Without knowing your age, weight, and talent, it’s just hard to say. Time goals are tough.
But 16 weeks with a good training plan will get you in shape and as fast as you are meant to be right now.
A year of base 30mpw, getting down to 12% body fat, followed by 12 weeks of a structured 5k speed plan could get you as fast as you are meant to be, or pretty close to it. Obviously 60mpw is better, but I’ll assume that isn’t in the cards.
You will know you are in range for a 20 min. 5k if your easy runs are comfortably around 8 to 8:30 per mile and you can comfortably settle into a 6:50 20 minute tempo run, which would feel great and smooth at first and be hard towards the end but still not feel like racing.