How and why did trade unions in America become influenced by organised crime, but unions in Britain didn't?

Trade unions in North America were often dominated by immigrants, e.g. Sicilians, and later Russians, who had been members of organized crime syndicates in their home country, prior to immigrating.

Trade unions in Britain were (mostly) dominated by "locals," who had no previous ties to organized crime.

I'll offer an alternative theory: organized crime in the US was to a great extent the product of Prohibition. It had given them immense profits, and a network of well-bribed police & other public officials, since many people regarded ignoring Prohibition as almost a public duty. Once it was repealed, the crime syndicates had to redirect those resources to other areas, like gambling (e.g. Las Vegas) and controlling labor unions. (Which, it might be argued, are at their extreme pretty close to an extortion racket anyway.) Britain didn't have Prohibition, so didn't have such large organized crime syndicates looking for new areas of endeavor.

I feel that answers linking crime to ethnicity should at least try to provide some documentation relative to the ethnicity of union ranks.

I would say that the factor more important was the use of organized crime (which existed before trade unions) to suppress worker movement.

Organized was a more important force in USA than in UK. In the USA, the organized crime intervened often in the laboral relations, with the clear example of the Pinkertons (that, while claiming to be "detectives", were often just band of thugs used to violently break strikes), initially to break strikes and teach "manners" to protesting workers.

In this aspect, once the leadership of a (once) demanding trade union was killed, jailed, or beaten enough to renounce their position, it just made sense replacing them with pliable men.

Of course, even with the mob controlling the trade unions, the workers need some improvents or there is always the risk of new trade unions being formed by disatisfied workers, so some improvements must be achieved.

And, in order to ensure that no other trade union could replace the old one (due to the ineffectivenes of latter), you could make union membership mandatory (which explains the creation of union shops and even closed shops in a country that declares itself "the paradise of free enterprise").

Additionally, trade union outside the protection of "controlled" trade unions continues to be very heavily punished¹.

Finally, once the mob controls the trade unions, it is not a great leap to search profit (more) from other opportunities that would arise from the situation.

In short, trade unions are corrupt because they were assaulted by the organized crime as a mean of forbidding representative trade unions from being formed/effectual. That is not news to the USA, many dictatorships (from Franco's Spain to those behind the Iron Curtain) had trade unions for exactly the same purpose.

¹: To illustrate my point, go to this category page in the Wikipedia and click links at random; check how many do include strikers being wounded or killed with nobody being held accountable.

NOTE: Yes I see the lack of citations, but the mob rarely makes their business agreements public (and when they do, there is a shocking lack of documental support). Given that, I understand (from what I have seen in previous answers) that there is a bit more leniency in the documentation part.

What is real and documented was the mob role (including Pinkertons) against the workers'rights movements, and what happened to the honest leaders that defied them. And well, the other answer include even less citations.

I don't know if there is a proven definitive answer to this, but let me propose an explanation from first principles.

At the time that Labour Unions were organizing in North America, and lobbying for legal recognition, careers such as teamsters and longshoremen were largely selected from the recently immigrated communities of Italians, Irish, and East Europeans. The tight knit nature of these immigrant communities provided an insularity from the community at large, particularly the legal authorities, that was fertile ground for organized crime.

In particular, the natural progression for immigrants was often to be low-skill manual labour the first generation, progressing into the police the next, and then into the more general middle class the third. This meant that for several decades around the turn of the twentieth century the ethnicity of the police and the low-end manual labour were different.

In Europe and Britain there was still a class distinction, but there would have been much more overlap with the class that populated the police, providing a much less fertile soil for organized crime to take root.