How I do politely indicate that I am expecting my name to be added to the paper I am composing this question on behalf of someone but as a first person. I am a postdoc at a top university in the US. I had recently been requested by my PI to help with setting up experiments and getting data for the latter part of a research that was to be submitted as a paper.

I spent 3 months aggressively working on it along side my own work towards other projects. The first author went ahead and submitted incomplete work without notifying any of us and without my name in it.

I had, thereafter, spoken to my PI and it looks like he is defensive of her submitting the paper because 'she was in stress and had to graduate. Not in the right frame of mind.' Between the two of us he said that he's going to insist on adding my name when they submit the revision (the paper already, as I expected, has request for data/corrections).

EDIT - I have indications that he is giving excuses and is currently passing the buck/lying to get the work done. Personally I have had no papers since I started this postdoc and I am afraid if I sound reluctant the PI will take away my work in my projects. This stresses me, but I also am not getting credit for where I worked. This is like a lose-lose situation for me.

The first author agrees to send the paper and review comments but doesn't. They expect real time work as though this is my only project. I am happy to spend time with some urgency if I at least get my recognition in the paper.

How do I word my emails or communications or speak to them and politely, yet clearly establish that this work from me has to be recognized in the form of addition to the paper?
If the situation is as you indicate and the original manuscript was submitted without the consent of all authors, then you have a case with the editor, although you should do your utmost to resolve this first with the thesis director.