Will Revert to You Soonest?

It means "Please wait and we will get back to you as soon as possible regarding your schedule". The use of "soonest" here sounds quite old-fashioned.

Will revert back to you soon?

~ "I will revert back to you shortly." The word 'revert' itself means to return to a previous subject or condition, so the insertion of the word 'back' in the sentence is incorrect. The correct thing to say is: ~ "I will revert to you shortly."

Will revert shortly?

The word revert means to 'return to a previous state'. This was the most frequently repeated email mistake i used to make. I used the word both as a an email response 'will revert shortly' and as a request "please revert asap" both are grammatically incorrect.