Cindy E. Harnett published an article in the Times Colonist titled "Latest word on levidrome: Oxford says it's not ready, but linguist begs to differ". In the article, she mentions that Oxford Dictionaries is not seeing a lot of natural use of levidrome.
The article consults an associate linguistics professor from the University of British Columbia. He states that natural use of levidrome is a tall order.
And we would agree.
When someone comes across a levidrome, or a palindrome, it is not as though s/he will actively tell the world about it. It would be more of a thought than a broadcast. It would be more of a "oh cool, that's a levidrome" - and life goes on.
In the article, it goes on by saying there are other proposed words with the same meaning as levidrome, such as antigram, heterodrome, or semordnilap. However these words haven't stuck. No websites dedicated to those words, unlike levidromes at https://www.levidromelist.com.
Natural use is going to be pretty hard for a somewhat obscure word in rarely used scenarios. Databases, which Oxford would use to determine word popularity and longevity, only work with written words or "published writings". So the more tweets or articles about levidromes the better. Spread the word.
So for now, levidrome ends up in the seldom used category with words like acnestis (part of your back you cannot scratch), meldrop (a drop of mucus at the nose) or octothorpe (the hash tag { # }).
The wait shall continue, and #levidrome (octothorpe levidrome).