You can't take the Attack action if your target isn't in range
When you take the Attack action you must immediately make an attack
When you take the Attack action you must immediately make an attack against a valid target.
As Jeremy Crawford has said, there is no gap between taking the Attack action and the attack that results from said action.
When you take an action, the action happens. There is no abstract "take an action" step that takes place before the action itself.
If you "take" the Attack action and don't attack, then you have not taken the action. The action and the resulting attack are one and the same.
You must pick a target as soon as you take the Attack action
Under the rules in the PHB under "Making an Attack" it lists the steps one must go through in order to perform an attack.
1. Choose a target. Pick a target within your attack's range: a creature, an object, or a location.
So, since the enemy is too far away to hit with your melee weapon you cannot choose them as a target for your Attack. However, if you didn't care about being able to attack the enemy, you could attack an object or a location that is within range and that would still be considered an attack for the purposes of Blade Flourish.
After you have completed your Attack, you can move (with extra movement)
No matter what you choose to attack, you can now move with whatever movement you have left plus 10 feet of bonus movement from Blade Flourish.
It is important to note that also that rules only allow you to move during an Attack action if you have multiple attacks. In that specific case, it only allows you to move between the attacks, but not before the first attack. No rule allows someone to move during an Attack action before the first attack.
However, if you are only able to make one attack per turn, then you just used your only attack on an object (for example) which gives you enough movement to reach them, but you have no attacks left to attack them with.
Scenario 1
The Bard can take the attack action to attack a rock nearby. After the attack completes, they can move 40 feet and be within melee range of the enemy. However, since they have already spent their Attack, they cannot attack the enemy this turn.
Scenario 2 & 3
As far as I can tell both of these are exactly the same for the rules that are relevant to Blade Flourish.
The Bard does not get extra movement unless they take the Attack action first. Since the enemy has total cover against the Bard, the Bard cannot attack them from their current position. They can attack an object or something and that will give them extra movement just like Scenario 1.
However, why not just use the Bard's normal movement to go around the corner? Once you see the enemy you can attack them if you are within range. If you attack the enemy (instead of an object/location) then you get the benefits of the extra movement and the extra flourishes the result when you hit a creature.
If the enemy more than 30 feet away however, then the Bard will have to choose between using their Attack to get the bonus movement (and not be able to attack the enemy) or if they want to take another action.
If all you want is extra movement, just use Dash
If you are already spending an action to gain movement and already losing your attack, then you should just take the Dash action and get a lot more movement from that spent action.
Really the true benefit from Blade Flourish comes from when you actually have something within your movement/range that you want to Attack and you get the movement as a bonus to that Attack. It doesn't seem to have much use as an ends to gain more movement when there is already an mechanic made for that very purpose.
Best Answer
Your speed will be 0 until the end of the turn
As quoted in the question, the description of the Steady Aim optional class feature for rogues says (TCoE, p. 52; emphasis mine):
This statement is setting the speed to an absolute value, and it defines the timeframe to be until the end of the turn. This is independent of the underlying speed. A wood elf barbarian with a base speed of 45 feet would be set to 0 by this feature, just as a halfling cleric with a speed of 25 feet would.
From a narrative/flavor perspective, the name of the feat implies that you stay put to get a better aim on your target. Page 62 of Tasha's Cauldron of Everything displays a drow rogue kneeling down to take aim with a crossbow (just below the description of Steady Aim).
The College of Swords bard's Blade Flourish feature description says, in part (XGtE, p. 15):
Blade Flourish increases your walking speed by 10 feet until the end of the turn, so if you had originally a walking speed of 30 feet, it now would be 40 feet. Then, due to the ongoing effect of Steady Aim, this speed is 0 feet.
D&D 5e has no formal stack which is used to apply and resolve effects in timing order.
Again, this also makes sense from a narrative perspective. Blade Flourish is intended to express "impressive displays of martial prowess and speed", the rogue doing somersaults and dashing about like in an Errol Flynn movie. This does not make sense with at the same time kneeling down and taking steady aim at someone.
To me it feels that other interpretations would be technical semantics to eke out a combat mechanical advantage from combining features against their spirit. Your DM may be down with that, and as in all adjudication cases, may give you pass to do so if you both have fun with it.
(There used to be guidance that you should consider combining any of the auxiliary works (Tasha's Cauldron of Everything, Xanathar's Guide to Everything, Volo's Guide to Monsters, etc.) with the core rules, but not several of them together, to avoid the otherwise unavoidable explosion of combinations that at some point will yield broken ones, but it is not official any more.)