I have never wanted to go for a run more than when I couldn't after my c/s!
If you were fit before and during your pregnancy then you really don't have to wait very long to get back into it. I only walked until my check up with DS1, but after getting the OK from the ob I started to increase the frequency/ intensity again. I was back playing soccer at 12 weeks post op.
Just a warning that sometimes you can feel really good while exercising but your scar will hurt afterward, so be careful not to go too full on too quickly. Make sure your body is comfortable.

I also didn't hold DS1 straight away. Many c/s mums don't. Try not to feel bad. As much as my vbac was an entirely different birth experience, there really is no difference in how much I love either of my boys, or how affectionate they are towards me. I am sure you are doing a wonderful job and that bond will develop over time.