Entity token availabe in Rules but non-functional
I'm not familiar with using the Button Field module in combination with the Rules module.

But from your question details, it seems that the entity:nid value is indeed blank. A possible solution is typically to add a Rules Action which is either of these:


"Fetch entity by ID" (if you know the ID you're processing).
"Fetch entity by Property" (if you have some criteria to build a list of things to be processed in a subsequent Rules Loop).


Especially the "Fetch entity by ID" is what I think you should give it a try. Also because of what's written in Comment nr 3 of issue nr 1796952, and which has some pseudo code like so:


event : on User clicks a button field
condition: Entity is of type on entity (make sure everything is in scope of rule)
actions: 


Fetch entity by id with ID (data selector) : entity:nid
pump the 'fetched entity' into a component and do things in there



If none of the above helps, then the "Submissions show up on reviewers profile page" part of your question is something that may help to use a completely different approach. If that's something like a Views result" (is it?), then you could maybe use either of these approaches:


Add a Rules Action to load a list the list of results created via a VBO enabled view.
Use the Views Rules module to have each row of that view processed via Rules, whereas you can make each column of such view available as a rules token, so if that "nid" is included in it, you have what you're looking for.


If none of the above help, You may want to use Rules debugging (eg to display values of relevant things using the Devel module, so that you can (temporary) add a Rules Action like "Display a value".
For more suggestions about that, refer to  "How can I display the value of a variable inside a condition?".

PS: those 2 AND Rules Conditions are not needed in your case (that's the default of how Rules combines multiole Rules Conditions).