What did some countries have that made them better equipped for nation building once they became independent? For example, the United States and India (both at some point British colonies) gained independence and reconstructed their nation with varying degrees of success. What contributed to this?
You're drifting out of history and into economics with this question. My source for this is Poor Economics and the freely available class on poverty economics that's available on MIT's website.
Speaking broadly, the success of a nation is determined by its government infrastructure. How well does the legal system, police, parliament, and other key systems of governance protect people and business. The basic theory from the book and course is that if the British, French, whoever is colonizing, were planning on actually living there then they developed a good infrastructure because they intended the place to be livable. Residents would want decent property and personal protections in place before moving there. Examples are the United States, South Africa, etc.
If they just wanted to strip the place of resources, they didn't develop anything except an effective government for removing goods. Colonizers don't really want the locals to have civil rights in that situation because it disrupts profit. So it would be a very top-down government with one or two leaders.
There are other cultural factors that vary from country to country, but generally speaking countries that are still struggling today were colonized and run like dictatorships. Switching over to democracy when your entire system is built for central authority is not easy.