It would reduce only the grappling character's speed, as the other answers explain it, but when taking it strictly RAW you can only use the grappling creatures movement to move the grappled creature: (emphasis mine)
Moving a Grappled Creature. When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
Basically, your steeds movement would end the grappled condition, because it would remove the grappled creature out of the grapplers reach.
The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.
A controlled mount cannot grapple
A grapple is a "special melee attack", it is a part of the Attack action. A controlled PC mount, despite being a wolf, cannot attack. It has only Dash, Disengage, and Dodge actions, see PH page 198 "Mounted combat":
It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge.
It can if it acts independently, but it still needs a free hand
Technically speaking, you can't give an order to grapple. But a mount can do things on its own. See PH "Controlling a mount":
While you're mounted, you have two options. You
can either control the mount or allow it to act
independently. Intelligent creatures, such as dragons,
The wolf still isn't eligible though, since it have to use a free hand, see PH page 195 "Grappling":
The target of your grapple must be no more than one
size larger than you, and it must be within your reach.
Using at least one free hand, you try to seize the target
A wolf has no hands and it has no actions in its stats block, allowing it to grapple without hands. Compare that with a Giant Octopus tentacle attack, explicitly saying:
Tentacles. Melee Weapon Attack: +5 to hit, reach 15 ft., one target. Hit: 10 (2d6 + 3) bludgeoning damage. If the target is a creature, it is grappled (escape DC 16)
No, you can not.
The grappling rules state that you can move a target you are grappling:
But, this applies only to you and your movement.
In D&D 5e, your movement is completely separate from the mount's movement. It even has its own turn distinct from yours; you can only control where it moves. Per the rules on controlling a mount:
Thus, the mount's movement will not allow you to move the grappled creature, and counts as an effect that removes the grappled creature from your reach: