There is no ё sound in English. I had to write my name either as Artyom or as Artem. Both are incorrect
Hmmm how interesting! Which did you settle for in the end? It's one of my favourite Russian names, it has a cool sound to it. Of course you could just make people call you "Art" which I think is a nickname for Arthur. In Scandinavia and Germany, the spelling is always Artjom. Assuming the person uses a rolling r, you get exactly the same pronunciation as in Russian.

I got tired of people abusing my name in English and simply adopted an English nickname. English people liked using the nickname, so it wored fine.

When I was travelling in some Russian speaking countries recently I was irritated that they routinely transcribed my name into Cyrillic letters as if it was an English name. So I got to hear the incorrect version of my name in English, pronunced with a Russian accent - that is too much. If I ever go to Russia for any extended period i will insist on the transliteration that I decide!