dnd 5e: Can you use Bladesong while you are wild and vice versa?

Looking at the subclass of Bladesinging Wizard, I realized that the Bladesong ability does not list any requirements outside of a Bonus Action.

From the second level, you can invoke a secret elven magic called Bladesong, as long as you don't wear medium or heavy armor or a shield. It adorns you with supernatural speed, agility and concentration. You can use a bonus action to start the Bladesong, which lasts 1 minute. He ends early if he is incapacitated, if he uses medium or heavy armor or a shield, or if he uses two hands to attack with a weapon. You can also discard the Bladesong at any time you choose (no action is required).

While your Bladesong is active, you get the following benefits:

  • You get a bonus to your AC equal to your Intelligence modifier (minimum of +1).
  • Your walking speed increases by 10 feet.
  • It has an advantage in Dexterity controls (Acrobatics).
  • You get a bonus to any constitution save cast you make to keep your focus on a spell. The bonus is equal to your Intelligence modifier (minimum of +1).

You can use this function twice.
You recover all spent uses when you finish a short or long

To my knowledge, there is nothing here or inside Wildshape that prevents you from using this ability.

I can see that it would be significantly advantageous to use this in a lunar druid construction, especially considering that it only requires a two-level fall.

Is there anything I don't know that prevents this from really working and that allows me to use both skills at the same time outside the need for 2 bonus actions? Use RAW, ERRATA or a clarification from 5E designers to explain why you cannot use these skills one after another.