Note 20 Ultra 5g Update

T-Mobile's firmware isn't the same as the Sprint firmware.  You will have to wait for the Sprint firmware update.  You could visit XDA Developers and check out how to convert the device to either the U1 firmware or the T-Mobile firmware, if that security update is important for you to have right now.  Otherwise, it should come at some point in the future but those updates all come from Samsung, so it won't roll out until Samsung gets it done.  

question..when you restart your phone does the Tmobile splash screen show up or Sprints?

Updates are always staggered.  Not all devices will get the update right away.  This is done to try amd prevent  major bugs from affecting too many devices.  

When I restart my phone - it doesn't say Sprint OR T-Mobile. The phone was purchased after the merger was complete. I am on the Feb 2021 firmware listed on the T-Mobile support page - https://www.t-mobile.com/support/devices/android/samsung-galaxy-note20-ultra-5g/software-updates-samsung-galaxy-note20-ultra-5g#steps2 

Just not the current version. 

Update!!!
I was able to get a T-Mobile SIM card and once I installed and restarted the update commenced. So now I am Sprint with an official T-Mobile SIM.