Update Unavailable with This Apple ID I've searched but cannot find the exact situation I have with Apple ID and App Store.

On my work iMac I have signed into App Store and downloaded software (El Capitan, Microsoft Remote Desktop to name a couple) with my personal Apple ID, but now I want to use a work Apple ID I have created. I cannot perform updates with my work account (as it didn't download the apps originally), is there a way of transferring these downloads from my personal account to my work account?
You can delete the Software in question, start the App-Store, logout the Work-id, login the Personal ID (in the App-Store) and Download the Software again.

If you want to make sure all settings and files of the Software in question are gone, use the Freeware 'AppCleaner' (https://freemacsoft.net/appcleaner/)