Seems to be some issues with the backend systems not rewarding players rank up items correctly, in most cases it is resolved by playing another round though. It is a known issue so at the latest it will be resolved once the bug is fixed.
The "next round" text is shown if you fulfill the rank requirements but do not own the item, with the assumption that that means that you will get it at the end of the round. Not the case since the system is a bit wonky.
We are currently limited to awarding items at the end of a round, or if you leave early and we know this is not ideal and it is being worked upon.