Animated objects as particles - Is there a way to tell them when to start their animations?

In order to do this you will have to make a gradient texture which starts from white and slowly reaches to black with circle emerging from its centre(can be easily achieved by using a color ramp.) Then you will have to unwrap the face you want to apply particles on and place it in centre of unwrap grid and then apply the texture in length of particles under textures tab.
For editing the materials you will have to use only one object to be rendered as hair instead of a collection( if you want variety of hairs you can add displacement modifier and change it to local for random results). After this you can go into material tab and animate the material(press I while hovering over your mouse on color). Later on in animation you can add random node in material or use multiple particle systems if you want to keep all the colours still there.