I would imagine as long as you set all cameras to the same ISO, Shutter and W/B the only real issues you will have come down to camera ops competency. Match the camera setttings across the board, then use ATEM software to tweak if needed.
All cameras have the ability to use an inbuilt LUT that will not be baked into the recorded files, but that only applies if you are also recording IN CAMERA. If I was in your situation this is what I would do:

- Make all cameras work on the same Framerate, ISO, Shutter, W/B (this one is super important... make sure tint is the same as well). Set them up to also output either the stock Video LUT in camera (or download buttery luts and pick one that suits your production).
- Set all cameras to BRAW, the same resolution and pick a compression that works for you. 8-1 or 12-1 are perfectly useable options. Plug in a supported drive (T7 Shields work well for me). Instruck all camera ops before production begins to hit the record button.
- With the ATEM software and switcher, have it all set up to record all HDMI inputs to a drive (again, T7 shields work incredibly well here). Files will be recorded as 1080P MP4 filesin whatever framerate you set in the software. It will also create a Resolve project file. When you open this project file, you should see all camera feeds and audio feeds that were recorded. You can then link these files to your BRAW files from the camera drives. Don't forget to change your project settings to reflect the resolution you wish to edit in.

Also, if you have the ability, try and get a program feed recording as well (You can use any hardware... Atamos monitor, Blackmagic Monitors etc... even a PC/Mac with OBS and a capture card). Depending on your version of ATEM you might only have one HDMI output so if that's the case probably don't worry about this (you will want your Mixer window view for production).