Although I am quite fluent in Russian, I wouldn't say English sound particularly 'cold' to me. There can be many different personal explanations on why we like or dislike certain languages: childhood imprinting, personal idiosyncrasy, aversion to or preferences towards certain languages - you name it.
But the psycholinguistic reason may be that Russian in fact offers a greater number of options of intonational patterns than English does. In Russian, the intonation actually conveys syntactical meaning covering the speakers's personal attitude towards the things said (or towards an addressee).
The difference may be also in MBTI-types of the actors (the Russian actors are more like Feeling types).
Also, please notice that while both stories cover the same archetypical scenarios (journey into a mirror, meeting doppelgängers and going through labyrinths), they are different and the Russian story is a bit secondary to Carrol's book, for it partially borrows the original idea (namely, travel through a mirror).
These ideas are well synthesised with Slavic archetypes of journeys to another world and were later developed in many Russian books.