It's clear how a creature both grappled and prone can stand - they can't. But what about the one doing the grapple?
Assuming both grappler (speed of 60 feet) and grappled are prone*, how does the grappler stand?
PHB pg 190/191
Standing up takes more effort; doing so costs an amount of movement equal to half your speed. For example, if your speed is 30 feet, you must spend 15 feet of movement to stand up.
PHB pg 195
Moving a Grappled Creature. When you move, you can drag or carry the grappled creature with you, but your speed is halved [..]
I can see three ways it can be interpreted:
The grappler has a speed of 60 feet. Standing up takes 30 feet. They stand, and only have 30 feet of movement left. Having someone grappled, they can only move 30 feet in any direction with the one they've grappled -- but they also have exactly 30 feet left. So they can move 30 feet in any direction with the one they grappled, so that being prone and grappling someone has no effect on you.
The grappler has a speed of 60 feet. Standing up takes 30 feet. They now have 30 feet left. Since they have someone else in a grapple, their speed is halved to 15 feet, and they can move with the grappled creature 15 feet.
The grappler has a speed of 60 feet. Since they have someone in a grapple, they only have 30 feet of movement. Standing up requires 30 feet of movement. So when they stand up, they have no movement left.
Also consider the case when the grappler has the Athletic feat, allowing them to stand up with only 5 feet of movement instead of half their total.
* They're both prone because the grappler jumped down onto an enemy, and grappled them as a bonus action via a Feat.