The context of the verse in question, is a discussion that arose concerning Jesus' prophecy about the future destruction of all the magnificent buildings to which one of his disciples had drawn his attention:
Seest thou these great buildings? there shall not be left one stone upon another, that shall not be thrown down.
-- Mark 13:2
Peter, James, John and Andrew asked him later:
Tell us, when shall these things be? and what shall be the sign when all these things shall be fulfilled?
-- Mark 13:3
Jesus' answer to their question included reference to:
- nation rising against nation.
- kingdom rising againagainst kingdom.
- councils and synagogues being places of corporal discipline for those who hold to Jesus' teaching.
- rulers and kings passing judgement on those who hold to Jesus' teaching.
It seems pretty clear to me that the "nations" Jesus mentions in Mark 13:10 are the same nations and kingdoms before whose rulers and kings his disciples will be brought, i.e. every sovereign domain - those with synagogues (Jewish), and those without (Gentile).
Conclusion
No. The text should not be changed because the translators have it right.