RAW: You can use Bladesong while in Wild Shape, and vice versa

You can enter a Bladesong either before entering Wild Shape or while in Wild Shape, as the Wild Shape feature states (emphasis mine):

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. However, you can't use any of your special senses, such as darkvision, unless your new form also has that sense.

Jeremy Crawford states in an unofficial tweet that:

Bladesong doesn't require a weapon in hand. Its description would say if it did.

And that a Bladesinger does not need to have his or her weapon in hand in order to utilize the Bladesong ability.

One thing to keep in mind is that you cannot maintain Bladesong if (per SCAG, p. 142):

you don medium or heavy armor or a shield, or if you use two hands to make an attack with a weapon

So keep that in mind when considering what to Wild Shape into.

You can use Bladesong while wildshaped, or can use it prior to wildshaping and still maintain its effect as long as you pick a shape that fits with the rules of using Bladesong.

This works but is exceptionally resource heavy and not particularly strong

The Druid's Wild Shape feature states:

You can use this feature twice. You regain expended uses when you finish a short or long rest.

The Spore Druid's Symbiotic Entity features states:

As an action, you can expend a use of your Wild Shape feature to awaken those spores... While this feature is active, you gain the following benefits:
Your melee weapon attacks deal an extra 1d6 poison damage to any target they hit.

This question ("Is an Unarmed Strike considered a Weapon attack?") and the Sage Advice Compendium both state:

For example, an unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.

So unarmed strikes will activate the additional 1d6 damage.

The Hexblade's Curse feature states:

As a bonus action, choose one creature you can see within 30 feet of you. The target is cursed for 1 minute. The curse ends early if the target dies, you die, or you are incapacitated. Until the curse ends, you gain the following benefits:
You gain a bonus to damage rolls against the cursed target. The bonus equals your proficiency bonus...
You can't use this feature again until you finish a short or long rest.

The Way of the Four Elements Monk's Fangs of the Fire Snake feature states:

When you use the Attack action on your turn, you can spend 1 ki point... [I]f you spend 1 ki point when the attack hits, it also deals an extra 1d10 fire damage.

All of these will work together but is requires a very lengthy setup and a huge amount of resources:

You will use up one of you two Wild Shapes.
You will use up your only use of Hexblade's Curse.
You will use up six of your 13 ki points.

It is nice however that all of these resources are recovered fully upon a short rest as well as a long rest. This is especially helpful with the spell catnap which allows you to take a 10 minute short rest (though it is a Wizard/Bard/Sorcerer spell).

Another thing to consider is the damage output:

Let us assume you instead had gone fully into Monk, used Flurry of Blows every turn, and buffed every attack with Fangs of the Fire Snakes

You would deal 1d10 + 5 (from dex) (magical) bludgeoning + 1d10 fire damage four times a turn for three turns. This would cost you 18 of your 20 ki and will deal an average of 192 damage.
However, the ki point expenditure here is less extreme because a 20th level Monk will have the Perfect Soul feature which states:

At 20th level, when you roll for initiative and have no ki points remaining, you regain 4 ki points.

Using your alternative method you will deal zero damage on turn 1.

On turn 2, you will make two unarmed strikes benefited by all three other effects. This will deal 1d8 + 5 (dex) bludgeoning + 1d6 poison + 1d6 Necrotic damage + 6 (proficiency) damage. Dealing an average of 22.5 damage per strike.

On turn 3, you will deal this damage again but add in the Fangs of the Fire Snake's 1d10 fire damage boosting the average up to 28 damage per strike and will be making four attacks.
This deal an average of 112 damage total.

Putting these together, across three turns, you'll deal an average damage of 157 damage, which is 35 damage less than if you were simply a level 20 Monk (though your method does have significant type coverage).

On top of all this there is two major problems: damage and death.

Your Symbiotic Entity bonus only lasts while you have the temporary hit points, and as you only have in the class you will only get 20 Temporary Hit Points.
The hex spell is also a concentration spell so each time you take damage the spell might end, in which case you will have top use up another bonus action to reapply it.

If the target of your Hexblade Curse dies before you deal all of this damage, the curse is gone, you cannot move it.
Additionally moving the spell hex to another target would use up your bonus action and thus cut down on your damage as well.
This tactic requires you to have three turn of setup, not take significant damage during those turns, and to have your target live through not only those turns but also your own attacks.

This is not a strong option as it has an extreme resource cost, has a lengthy set-up when combats in 5e-dnd are normally quite short, and multiple parts of the plan are hindered by both damage and death.