Practically wherever the british and irish went in numbers of a few millions, the countries flourished.I can think of Canada, USA, AUS+NZ, and Hong Kong. I would rekon that the pursue of free market/ rule of law/ property rights and freedom of entrenorial initiative drove their ascension. What set the apart form the latin america for example, where large numbers of europeans also immigrated?
For your hypothesis to make sense you would have a lot of work to do. First of all you would have to define what constitutes "successful", then prove that this was specifically due to immigration/colonization. After that, you'd have to prove that it's a uniquely British/Irish phenomenon. Then you'd have to exclude the possibility that it's simply a statistical conincidence given that Britain colonized about a quarter of the entire planet and 20% of the entire world population by 1922.
When you've done that you'd need to account for other factors of your hypothesis such as Hong Kong never having "a few million" British/Irish Immigrants (in 1861 there were less than 2000 Europeans/Americans in a population of 125 000). You would also have to ensure that no nation colonized by another colonial power amounts to whichever standards you have defined as "successful" or at the very least explain why this was the case.
When you've done that, we can start looking into why it happened. I expect we'd find a stronger connection to Eurasian majority populations than to specific british/irish populations, it'd also expect that it can be explained by roughly the same model as the survival of Eurasian civilization on the world stage. Guns, Germs and Steel.