I went through the exact same thing a few months ago. According to the letter of the law you are supposed to re-apply immediately if your company goes through this type of re-structuring. However I didn't and waited until my current
TN was about to expire with Company A and then went to the border with my letter stating that Company A had re-structured and spun off Company B and I had been transferred to that company.
At the border I even told the officer that we had re-structured when he asked if the TN was from the same company. Then on my I-94 form he just wrote Company A/Company B. Not sure if this is normal but clearly in my case they didn't see any issue with the change. I think financial or corporate re-structuring are so common that they are just used to them.
If I was to do it again I would probably just re-apply the next time I crossed the border just to be sure it never came back to haunt me. (My paychecks started coming from Company B on Jan 1 and I renewed in mid Feb)