How do I see if I am top or bottom side in the new client?

I have no idea. I am writing now to mention that even the Gem above the nexus which indicates your side appears to be missing for my last few games as well so there is no visual representation for your side, or role. At least you can read which role you are.

EDIT: I misread and I thought you meant which role. My bad. Either way, there should be a illuminated gem above which nexus will be yours when the mini map of the rift is up.

When the lobby loads it shows a map with which side you're on. Plus Blue side bans/picks first, so you can track that.