When will we get a dragon rider subclass?

Make it a series of feats with prerequisites that anyone can pick up. 

This would make a much more interesting world.

Different classes can ultimately gain different Dragons as friends/mounts. Never dominating the sapient Dragon. Other dragons would not like that, in fact they could view it as slavery.

You would have to have an insentient species just like other mounts. And nurf the mounts power. Breath weapons and flight for low level characters are very powerful.

A hundred ideas could be added to your world. But like others have said. Make sure the DM is not handing a player the cheat code.

The point about being non-sentient- or at least not previously extant beings who would have lives and goals outside of obeying your instructions- is covered by Drakewarden. The issue is that despite recent pushes for "setting neutral" content and a downgrade to the emphasis of lore, you can't meaningfully have a "real" dragon by the reckoning of most D&D players that also fits the mold of a subclass feature companion.

Can dragon-riders be good to explore underdark?

Maybe a dragonsteed-rider and this with climb speed could be useful to explore underground.

It might be super easy to figure out, but you certainly haven't. If you think a flying mount a level 5 or 7 is breaking the game, then you have no idea what base classes and existing subclasses can already do.

I have likened the dragon to paladin's "find steed" which can do the same thing, to sorcerer's "summon draconic spirit" which is substantially stronger, and there are easily craft able flying boots, carpets, brooms and many other ways to fly, even without concentration.

If you can't balance for stronger Drakewarden, you can't balance against half the classes.

To be honest, I would be ok with drake instead of a true dragon if it was stronger than before. But I also think dragon doesn't have to be such a problem. As I've said, the dragon can act more like warlock patron. They are usually crazy powerful and it doesn't make all adventures pointless. Have the dragon only start to grow into their power early on and act like a familiar, maybe even a non permanent one. Then it can be a remote presence that talks to you in the same way patron does (magically or through occasional visits) and you call call them once a day, or several times a day where they fly/teleport in and you can ride them properly. This impermanence would solve the role problem, they have their own stuff going on so they show up only when you really need them. It would also mechanically differentiate the ranger or fighter from other pet subclasses. And everyone thinks patrons are a great tool for DMs so having another class that's not a warlock with essentialy the same thing would allow for more plot hooks.

I've already said this elsewhere, but I'll reiterate. I'm not saying the dragon is supposed to be enslaved or anything similar. More like a warlock patron which instead of eldritch blast and 9th level spells gives you their help as a mount and partner in a limited fashion. 

You can give that to anyone. But it seems a shame to miss this opportunity to give classes that much need extra flavor and cool stuff, something cool and unique. Wizards don't need a dragon on top of what they already have. But fighters are desperately plain in my opinion, rangers are close behind. I guess figgters got better in 5.5, but still. Why not give them a bit more?

I've addressed this in like three separate comments. I will try to add an edit to the original post to clarify. It is not supposed to work exactly like a pet. More like warlock patron.

That's one of the arguments that are actually in favor of true dragon instead of a drake. Dragons can shape shift and they'd propably be only called in sometimes. Not a permanent companion. So eve if you're in a cramped space, they could appear as a humanoid caster summon, or they move through a corridor as a humanoid and turn into their real form once they get into an open cavern or room. That way you could use them in any situation.

You repeat “it would be like a patron” as if that solves all the problems.  It does not. Frankly, it probably creates more of them.

For starters, patrons are typically NPCs under the control of the DM, while the warlock’s powers are something the player controls.  That clearly defines the roles of each party and keeps there from bleeding lines between player and DM roles.  This is what makes them useful DM tools - they are an NPC that can always int react with the party when needed.

Where your idea breaks down is now you have the "patron” dragon a physical part of the party.  Now the line is blurred.  If it is to serve as a patron, then the Dm is to control it… but if it is a core set of mechanics, then the player does.  Introducing an NPC that is simultaneously both patron and pet who different people controlI at different points creates a party dynamic problem.

You also just have a problem with nonsensical situations becoming likely.  Let’s say the party is in a hard fight. Death looks possible. You have a situation where the dragon would effectively be saying “I am more powerful than this, but you have not unlocked the ability to use more of me. Guess we are all dying because, even though I am in the room right now, you don’t get these mechanics yet.”  Kind of takes some of the verisimilitude away if your “patron” is in the room and dies because they won’t use their whole strength.

Patrons work when they are distant figures conferring power.  They do not work so well as a joint PC/DM NPC who is also a frontline combatant in the party.

True dragons are far too powerful to be in a party. Thats it there is no argument you could make that would let them be in a party.
Flight and Breath weapons are far to powerful to hand to a PC. Even if its a nurfed very young dragon, mama will NOT let that happen.

A non intelligent Drake maybe. Not much of a breath weapon and very limited flight and carry capacity.

As a PC with a flying steed all I needed was the spell/item Feather Fall and I became an airborne assault force.
Tell the steed to go hide and I would just fall off dropping well behind enemy lines. Cast the FF spell/item on the way down and always land safe.
A few weeks of practice and I could guide my way to a specific roof in a town or castle.
Add in a few one gallon jugs of alchemist fire and you could get the idea of just how powerful a low level PC could get with just the flight ability.

Getting a dragon steed is normally something a PC would retire with as soon as acquiring. Your life long goal.