Can you use Bladesong while you’re Wildshaped and Vice-a-Versa?

Looking over the Bladesinging Wizard Subclass, I came to realize that the ability Bladesong doesn’t list any requirements outside of a Bonus Action.

“Starting at 2nd level, you can invoke a secret elven magic called the Bladesong, provided that you aren’t wearing medium or heavy armor or using a shield. It graces you with supernatural speed, agility, and focus. You can use a bonus action to start the Bladesong, which lasts for 1 minute. It ends early if you are incapacitated, if you don medium or heavy armor or a shield, or if you use two hands to make an attack with a weapon. You can also dismiss the Bladesong at any time you choose (no action required). While your Bladesong is active, you gain the following benefits: • You gain a bonus to your AC equal to your Intelligence modifier (minimum of +1). • Your walking speed increases by 10 feet. • You have advantage on Dexterity (Acrobatics) checks. • You gain a bonus to any Constitution saving throw you make to maintain your concentration on a spell. The bonus equals your Intelligence modifier (minimum of +1). You can use this feature twice. You regain all expended uses of it when you finish a short or long rest.”

To my knowledge there’s nothing here or within Wildshape that prevents you from using this ability.

I can see that it would be significantly advantageous to use this in a Moon Druid Build, especially considering it only requires a two level dip.

Is there something I don’t know preventing this from actually working and allowing me to use both abilities at the same time outside of the need of 2 bonus actions? Please use RAW, ERRATA, or a clarification from the designers of 5E to explain why you couldn’t use these abilities one after another.