Two ways to approach this.
1. Use an expression to offset the effective start time of the animation. Apply this expression to any animated properties in your first layer. Then, when you copy and paste it onto layers with different start times, Ae will play the animation when the layer begins, irrespective of when the keyframes occur on the time.
if (numKeys < 1) { value; } else {
var animationStartTime = key(1).time;
var animationStartDelay = thisLayer.inPoint - animationStartTime;
valueAtTime(time - animationStartDelay);
}
2. You can quickly move all layers to the beginning of the comp by selecting them all, placing the current time indicator at the very first frame, and pressing the ‘[‘ key. Then, you can copy and paste and everything will line up at the first frame. To re-sequence the layers, select them all (first to last), then in the timeline window, right-click any of the layer bars, select Keyframe Assistant > Sequence layers…, and then click OK in Sequence Layers dialog (keeping ‘Overlay’ disabled).