Hello ! (sorry for my bad english)
There was simple mod i was using in Civ5 that was disabling units movement animation after you clicked "next turn", and re-enabled it at the begining of next turn. Units movement animations always been a cause of long waiting time between turns, and it got even worse when they introduced trade caravans in brave new world.
I tried it in CivBE and it still work pretty well. (although it doesn't show which sponsor/alien turn is being played, no idea why). In my case, at late game (turn 150+), time between turn is more than halved, which is fantastic.
I am not the author. You can find the mod in this topic (8th post on the first page has a link to download it): http://forums.civfanatics.com/showthread.php?t=503869
You just have to rename it from "QMMOD (v 1).civ5mod" to "QMMOD (v 1).civBEmod" and drop it in your mod folder. (renaming is required for CivBE to recognise the mod).
I'm sure it could be improved to be "fully" compatible with CivBE, but for me it work pretty well as it is.
EDIT: Don't need a new game, it works in existing saves
There was simple mod i was using in Civ5 that was disabling units movement animation after you clicked "next turn", and re-enabled it at the begining of next turn. Units movement animations always been a cause of long waiting time between turns, and it got even worse when they introduced trade caravans in brave new world.
I tried it in CivBE and it still work pretty well. (although it doesn't show which sponsor/alien turn is being played, no idea why). In my case, at late game (turn 150+), time between turn is more than halved, which is fantastic.

I am not the author. You can find the mod in this topic (8th post on the first page has a link to download it): http://forums.civfanatics.com/showthread.php?t=503869
You just have to rename it from "QMMOD (v 1).civ5mod" to "QMMOD (v 1).civBEmod" and drop it in your mod folder. (renaming is required for CivBE to recognise the mod).
I'm sure it could be improved to be "fully" compatible with CivBE, but for me it work pretty well as it is.

EDIT: Don't need a new game, it works in existing saves