You need to use "than" with "no sooner" sentences. Even with "than" added, your example sentence is a little bit strange. However, it wouldn't take much to fix it. It's also important to note that the quote probably comes from the Russians, because they refer to "us" at the end. No sooner do our colleagues in Europe or the US drive the situation into a dead end, than they say that Moscow holds the keys to a solution and put all the responsibility on us. I added "than", removed "always". "than" is needed. The use of "always" is very awkward because usually "no sooner... than..." doesn't refer to a regularly recurring situation. No sooner does A happen, than B happens. (O) No sooner does A happen, than B always happens. (X)

Your sentence is hard to understand because it is poorly written. It says "No sooner do our colleagues in Europe or the US drive the situation into a dead end, they always say that Moscow holds the keys to a solution and put all the responsibility on us." However, it is not clear just what the author means. He/She probably means something like "Our colleagues in Europe or the US are always bringing negotiations to a standstill by claiming that Moscow holds the key to a solution, thereby placing the onus of responsibility on us." It is not good English to use "no sooner" without "than". Hope that helps.

The sentence is quite correct. The grammar is called "inversion", and is a result of putting "no sooner" at the beginning of the sentence. You can understand the sentence by replacing "no sooner do" with "as soon as". Basically, as soon as one thing happens, another thing immediately follows.