The problem is likely caused because your browser is caching the old image. Try doing a hard refresh with control+F5 on your Chess.com homepage and you should see the update.
If this does not work possibly try closing out the browser and re-opening it or logging out of your Chess.com account and logging back in.
I uploaded a Avatar I like and it shows up everywhere except Live Chess, whats up with that? Do other players see their own avatars?