Most former colonies are now developing or poor countries, such as many countries in Latin America, Asia, and Africa. Some exceptions are Australia, the United States, Canada. Interestingly they are mostly populated by immigrants from Europe. Historically, was their being countries of European immigrants in some way related to their development to become prosperous, first-world countries?
If you'll allow me to be a little crass here, I think the meat of your question boils down to, "This looks racial. Is it?" I'm gonna say, No.
I think it would be more comprehensive (and perhaps accurate) to say the more successful former colonies are mostly populated by direct descendents of the colonizers, rather than the indigenous population at the time of colonization.
Not only does this remove the racial element, it allows you to explain successful societies like ancient Carthage (founded and peopled by the Semetic Phonecians), and the various successful Greek colonies of the ancient era (the Greeks were "white" Indo-Europeans, but so were most of the natives). These all shared some similiar charactaristics with the British colonization efforts you mentioned: they were efforts to build home-country style societies on the foreign shore, with no input whatsoever from the natives, and they served as a valuable release valve for overpopulation or persecution problems at home.
Contrast this with the less wealthy former colonies in the New World, like most of the former French and Spanish colonies. In their case, the colonial drive was mostly economic, not demographic. So rather than rebuilding their home societies with their own people, the colonizers just sent over the bare minimum amount of their citizens required to exploit the resources of the colony.
Its like the difference between a mine and a home.
Singapore can be considered a first world country today and is not politically/culturally influenced by European immigrants. Singapore today is mostly descended by Chinese immigrant labor as well as the local Malays. The main reason for their modern development and security is their alliance with US/Israel, as well as their strategic location.
Malaysia, while slightly less developed than Singapore, had even put in place an indigenous protectionist policy to prevent the country from being politically controlled by immigrants.
T.E.D is quite spot on the reason for this: It's because of the development brought in by colonists. The indigenous population often have less ambition and less technology. British colonies were built to be homes. They brought in infrastructure, cash crops, advanced mining technology. When colonists gave independence, the former colonists had a strong economy and a ready market (their former colonists) to grow into a powerful economy.
Another key factor would be in having ambitious, sincere, highly educated leaders. It's often unlikely for these leaders to be of the indigenous population to strengthen control over the locals. However, examples like Abdul Razak and Lee Kuan Yew (both lawyers educated in Britain), show that race doesn't matter as much as education and motivation.
Jack Rakove has splendid series of lectures on itunes that expands on @Samuel Russell's distinction between "resource extraction" and "Settler" societies, and compares the approach of French and English settler societies. The lectures hint at much deeper and broader scholarship, but (a) it is difficult to cite a podcast as precisely as a literary source (b) he is providing context prior to the real meat of the lecture and (c) it has been a few years since I listened to them. With those caveats in mind, those lectures and the associated course material may be helpful in further researching the answer.
But IMHO the answer is tea. Colonies succeed if they are endowed by their parent empire with the habit of drinking tea.