Will external macOS drives work on a new Mac Studio?

They'll remain as is as long as the OS on them is at least as new as the version which originally shipped with the replacement computer.

(262930)

Another factor that makes this more difficult is that a modern (post 2018) Intel Mac will not boot from any external drive until you make adjustments to its Privacy & Security settings to authorize that.

About Startup Security Utility on a Mac with the Apple T2 Security Chip

https://support.apple.com/en-us/102522

a Mac with Apple-silicon may force you to re-install a new instance of MacOS on that drive before it can create the required locked system snapshot:

How to use an external storage device as a Mac startup disk

https://support.apple.com/en-us/111336

bradch_00 wrote:

The OS versions on the external drives are newer than the 2022, I think Ventura or Monterey, version, one is Sonoma and the other Sequoia as I was running Tahoe on the internal drive.

A brand new Mac Studio, today, will come with macOS Tahoe preinstalled. You will not be able to boot it from a drive with any earlier version of macOS ... that includes Sequoia, Sonoma, Ventura, Monterey, etc. Nor can you downgrade it to a previous version of macOS.

'What version can be installed' is a very confusing issue, but these subtle rules apply:

A version of MacOS that is OLDER than the original-release software for a particular model Mac cannot possibly run on that Mac, because the newest drivers for its new hardware are not included. (Also, you must use release numbers, like 17E56 not just revision numbers 10.13.2 to make a completely accurate assessment.)

The instance of MacOS that "shipped in the box" with a specific computer likely works with that model only -- it does not have "drivers for every appropriate model Mac" so will not run on different models. Any updates you perform on that MacOS software, including re-install of the same major version, may still have only drivers for that model Mac.

bradch_00 wrote:

The power supply and the main board have corrosion on them (probably from being on constantly, with the work I do, I rarely turn computers off, lesson learned perhaps)

If there is actual corrosion on the metal & electronics, then that means either a liquid spill, or using the computer in an environment with very high humidity, or maybe operating the computer too close to an air conditioning duct. Running a computer 24/7 won't cause corrosion by itself.

My question is, will the external drives with Mac OS on them need to be rebuilt from scratch, or if I choose one will it reboot and be useable after some configuration and security information is (most likely) entered?

It is hard to say.

Did you actually install macOS onto these external drives? Or did you clone them? The former may work given the caveats provided by the others. The latter is less likely since I've seen people report a cloned drive may not always work on the same device after an update.

Be very cautious going forward with that older Mac Studio. Corrosion is a problem that tends to 'creep' and get worse over time, and that Mac could fail suddenly. Maintain frequent backups and be ready for it to fail without notice.