By my 4th child, I was packing a gym bag for a diaper bag!
I had 4 in 6 years, so they were petty close together. I broke it down into two bags. One never left the car, it had changes of clothes, diapers, formula, meds and any possible thing I could think of that I might need. Then I had a small over the should bag that had basics for stopping at a friends house, or going to church.
Just a few ideas in regards to the prepping situation. I think cloth diapers are great, but if you are in a SHTF scenario, where you might have to cut and run, disposables might be better. They hold more and last longer when wet, and you don't have to worry about where to wash out a cloth one. Perhaps if you just stored away a couple boxes of cheap diapers (if there is such a thing) for emergencies.
As several others have mentioned, keep baby advil/tylenol handy. Also ask your pediatrician for a med dosing weight chart. That way if you are not able to get a hold of your doc, you will know how much medicine to give. Another important one to remember in my opinion, is benedryl. You never know if your child will have allergies or an allergic reaction until it happens. My 5 year old has had several severe reactions over the years, the first at 9 months, that we didn't see coming. Even now, we never know what might set her off and make her face swell shut! I keep bottles of benedryl stashed everywhere!
Also, screw top tubes of ointments, like desitin or A & D ointment, travel better in the car. They are less likely to leak when they get warm. I still store them in a ziplock bag, just in case.
I always kept a bottle of water in my diaper bag, incase I didn't have access to fresh when I need to make a bottle. As they get older, I added travel size boxes of cheerios and things like that. They travel well and last for quite a while.
Many blessing on you and your precious addition. When my first was born, my biggest joy was just sitting with her across my chest and smelling her hair. It was the ultimate contentment. Enjoy every moment.
