Scenario 1: The mount is charmed.
If the mount is charmed, then it will probably fall. From the section on Flying Movement in the Combat chapter (PHB 191):
If a flying
creature is knocked prone, has its speed reduced to 0, or
is otherwise deprived of the ability to move, the creature
falls, unless it has the ability to hover or it is being held
aloft by magic, such as by the fly spell.
Hypnotic Pattern sets affected creatures' movement to 0, so unless the mount can hover or is being held aloft by magic, it will fall.
It's unclear how long falling takes, so depending on your DM, the rider may or may not get a turn before they hit the ground. If they do, they can use their action to shake their mount out of its stupor as described in the spell. If not, their mount will fall and hit the ground.
If the mount can hover, or is held up by magic, then they still won't be able to do anything, but they won't fall, either.
Scenario 2: The rider is charmed.
If the rider is charmed, things get a little ambiguous. I won't post the entire text of the Controlling a Mount section here for copyright reasons, but suffice to say, it doesn't actually mention under what circumstances a rider can or can't control their mount.
If your DM decides that the rider can control their mount while incapacitated (seems unlikely, but you never know), then the rider will still lose their own turn but can (somehow) direct their mount on its turn.
If your DM decides that the rider can't control their mount while incapacitated, then the mount will act independently as described in the Controlling a Mount section. In this case, the rider will either be sitting there while the mount does whatever it wants, or possibly be thrown off. This is all pretty much up to your DM to decide.
Scenario 3: Both rider and mount are charmed.
As in scenario 1, if the mount can hover or is held up by magic, it won't fall, so rider and mount will stay exactly where they are. But if not, there's nothing either of them can do except fall helplessly to the ground.
There's no easy way.
Incapacitated doesn't mean helpless, or that a creature's defenses are down.
Incapacitated creatures are aware of what's happening around them, can move normally and interact with objects, and can oppose attempts to move, grapple or disarm them.
Grabbing an incapacitated creature still requires a Strength(Athletics) check opposed by the creature's Strength(Athletics) or Dexterity(Acrobatics). Depriving them of something they are holding would also be resisted by their Strength(Athletics).
Incapacitated creatures aren't especially vulnerable to intimidation, persuasion, or anything else. Their AC isn't lowered. And so on.
The Hypnotic Gaze feature, however, has the specific additional effect of also charming the affected creature; that means that some interpersonal interactions can be attempted (by the enchanting enchanter) with advantage. But that also wears off as soon as any damage is dealt.
Best Answer
Hypnotic Pattern has no impact on a saving throw whatsoever. It imposes both the Charmed and Incapacitated conditions and having a speed of 0 does not automatically impose the immobilized, paralyzed, restrained, or stunned conditions, it just means you have a speed of 0. In fact, immobilized isn't even a condition in 5e.
Rules for saving throws can be found on page 179 of the Player's Handbook and nowhere does it say that they use either an action or a reaction, it simply says they "represent an attempt to resist a spell, a trap, a poison, a disease, or a similar threat" so being Incapacitated has no effect on this.
Being charmed means you can't attack or target the charmer with harmful effects but in no way affects your ability to "resist a spell, a trap, a poison, a disease, or a similar threat".
Conditions and spells that do affect savings throws explicitly note this fact in their descriptions.