A wizard is fighting with his bladesong active and decides to cast polymorph upon himself. Will bladesong still be active in the new form (let’s take a giant ape for example)?
Yes, class features like Bladesong are cancelled under polymorph
Polymorph is very specific in what it's doing:
The target's game statistics, including mental ability scores, are replaced by the statistics of the chosen beast. It retains it's alignment and personality.
Since your stat block has been completely replaced by the new creature (except for alignment and personality), you no longer have access to your class features, including Bladesong. Anything that you were using is no longer happening because you simply don't have access to it.
Bladesong also requires the creature to be an elf in order to even access the feature. Once a beast, you are not only not a wizard, but not an elf.
A case where it would carry over
Do note that Wildshape does allow for you to carry over and is the specific beats general case:
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.
A similar case considered by Jeremy Crawford
A similar question regarding Rage and Polymorph was answered by Jeremy Crawford who supports this as well:
Polymorph replaces your game statistics, including class features, with those of the beast. If you're a barbarian, you lose Rage.
Both of these are class features, both involve the creature being polymorphed. Both cases should be the same in terms of final effect, which is loss of class feature.
Crawford contradicts himself
As can be seen in this tweet Crawford has a very different ruling about Arcane Ward and Polymorph.
If you're protected by Arcane Ward, changing your form doesn't end the ward, unless the transformation effect says otherwise
This is a problem with relying on Crawford and Twitter in that there may be conflicting answers and we're once again left to figure it out ourselves and decide which direction makes sense.
Looking at the words of the spell polymorph and the restrictions around Bladesinging, it doesn't seem to make a lot of sense for it to continue while a beast.
There is no reason for Bladesong to end.
Bladesong is an effect with a duration, and special termination rules. It is canceled when the user wields a weapon in two hands or puts on medium or heavy armor. Polymorph is an effect with certain disadvantages, including starkly limiting (and altering) the available actions of the target. Polymorph does not cause you to wear heavy armor or wield a weapon in two hands, however, and bladesong by RAW does not require further action (and also does not require you to use a weapon at all). There's nothing causing the effect to end, so by default it continues (as silly as that is).
Given the fluff, however, it would be entirely reasonable for a DM to conclude that it ought to terminate (how do you use bladesong with no blade?) and houserule it. If I were DMing a game and the issue came up, I'm not sure which way I would rule.