I am wondering if there is any nuance or difference between them ?
Yes, there is. I'll try to explain my opinion.
I am also wondering if there is some politeness attached to it ?
No, politeness isn't the factor
- I would say "along with"
- Timing is not important compared to 同時に
- Larger scale compared to 同時に
- Dynamic and impactful compared to 同時に
- At a concert, 一斉に拍手 round of applause
- At a marathon, 一斉に走り出した start running at once
It's used when a phenomenon occurs "along with" each other. It's not really important that we start clapping our hands at 10:00:01, is it?
- I would say "simultaneously"
- Timing is important compared to 一斉に. I say compared to 一斉に, because 同時に is not always used for timing-critical stuff
- Not necessarily a large-scale or group activity, can be used for small boring things (press button simultaneously)