Honest answer: sooner than you fear, but maybe not as fast as you'd like.
Most beginners feel a real shift in three to four weeks of consistent running — runs that felt brutal start to feel merely hard, then manageable. Your heart, lungs and legs all adapt, just at slightly different rates, which is why some weeks feel like a leap and others feel flat.
The catch is "consistent." Two runs a week, every week, will get you there. A heroic week followed by ten days off resets the clock. It rarely feels linear, but it does keep moving — keep going and the easier days arrive.