I'm having the exact same problem. I tried the steps given (unsubscribe, delete mod folder...etc), redownloaded and ensured that the ME mod was the only one selected (green check mark).
When I set up new game and look at races...only the old one are present, none from ME.

I'm sure I am just doing something wrong, but if someone could help that would be great! Thanks

I'm an idiot - figured it out.

When you enable the mod, hit "Single Player" a the top of the active window to configure the mod. I was hitting "Back" assuming the mod was enabled and I needed to go back and set up a game.
Hope you're having the same problem so I'm not the only one making this mistake...

I have clicked the "Single Player", then the "SET UP GAME", but there were still no ME civilizations. Can someone please tell me, what am i doing wrong?

Try the step-by-step guide in the troubleshooting.

OK, so I now know what is the problem - I don't have the "Brave New World" DLC, I thought it was included in the GOTY version of the game, but it seems it's not. Thank you for the advice anyway.