The original plotline for Chess involved not one, but two, matches -- one between Freddie and Anatoly, and the second between Anatoly and a Russian who was later named Leonid Viigand. Viigand is a nonentity who is most noteworthy for the fact that he spends the entire vodka-soaked dance sequence of "The Soviet Machine" practicing his chess moves. The rationale for Viigand was to have a parallel match to the 1978 Karpov-Korchnoi match, as the first was a parallel to Spassky-Fischer 1972. It was followed through for London, but not resurrected for Broadway, Sydney, the American Tour, or Stockholm.
As a fundamental matter of dramatic principle, the split-match format is the key weakness of the London show. For their flaws (all of them have second-act problems), the later variations have a much greater coherency, which stems from the fact that the single chess match acts as a framing event for the dramatic action. Here is the essence of the split-match problem.
If a split-match format (Anatoly/Freddie, Anatoly/Viigand) is used, on the one hand, the stakes in "Endgame" are high, because Molokov's goal is to make Anatoly lose the match. On the other hand, in the time between the second act's opening and "Endgame," it is very difficult to make this match matter to Anatoly enough for his winning despite the cost in "Endgame" to make sense.
If a single-match format (Anatoly/Freddie) is used, the stakes in "Endgame" are lowered, since Molokov's main goal is to get Anatoly back to the Soviet Union. This makes it possible for Anatoly to lose the match, and in Stockholm "Ni dömer mig" ("Endgame") is not even used for the final game. It is also harder to place "One Night in Bangkok," which was a problem in Broadway and the American Tour, and which Stockholm punted on completely. The benefit of a single-match format is that the action is much more cohesive, with the match providing the outline for the drama.
Historically, it is probably more popular for Anatoly to win the final game than Freddie. Critics of the Broadway version point out that in New York, the American won, but really, this has less to do with nationality and more with characterization. The thrust of the concept album was that winning mattered to Anatoly more than the secret about Florence's father; this comes out in "Argument" and is reinforced throughout "Endgame." However, the London Anatoly has less of this aspect to his personality, and his "Endgame" material comes out of the blue. By the time we reach Broadway, a more fatalistic Anatoly losing the match makes sense.
More than anything, the split-match format exposes the main contradiction in Chess: although Anatoly is the person who makes the main decision in the show, Florence is its heart and soul. There are traces in the concept album that Anatoly was originally the center of the musical, but it is not so in the actual productions. So, the decision to weight the outcome of the second match goes against the grain, and London's ending does not have the intended impact.
Since Stockholm did not return to the split-match format, it is more or less likely that any English-language revival will follow suit. As a whole, the dramatic advantages of the single-match tend to win out, and barring significant recreation of Anatoly's character, it is best to go with the flow of the post-1986 productions here.