I am submitting this answer in order to provide a definitive answer to this question, in light of the submission by @Phil S.
The Answer
This answer is functionally identical to the answer provided by @SevenSidedDie, which is marked as accepted.
Your grappler cannot move after standing up without ending the grapple.
When using the Athletic feat, 5 feet are deducted from your movement, and you can then move as usual under these conditions.
Definitions
Firstly, some definitions to clear up a misconception that has taken hold of this thread.
Your Speed refers to the value on the character sheet:
[...] Your speed - sometimes called your walking speed - is noted on your character sheet.
(PHB, pg. 189, "Your Turn")
Your Movement is not subject to a simple definition, but it is generally used to indicate the distance you can walk in your turn. (See: PHB, pg. 190, "Breaking Up Your Move")
Whenever you move, you subtract the distance travelled from your movement, keeping relevant modifiers in mind (such as difficult terrain).
Step By Step
So, to use the example:
- The grappler, when attempting to stand up invokes the relevant rules:
[...] Standing up takes more effort; doing so costs an amount of movement equal to half your speed.
(PHB, pg. 190 - 191, "Being Prone")
The grappler Does not invoke the rules for moving a grappled creature (yet), as the grappled creature is not being dragged or carried - it's not being moved at all.
This leaves our grappler with \$60 - 30 = 30\$ft of movement
- Now, the grappler wants to drag the creature and invokes the grappling rules:
[...] 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.
(PHB, pg. 195, "Grappling")
Thus, the grapplers speed is halved, resulting in \$60 - 30 = 30\$ft of speed, and we run into the following rule:
[...] On your turn, you can move a distance up to your speed.
(PHB, pg. 190, "Movement and Position")
Since the grapplers speed is 30ft when dragging a grappled creature, and 30ft of movement has already been expended, the grappler is unable to move while dragging the creature. But, they can still release the grapple and use the remaining 30ft of movement, as the grapplers walking speed is then not halved.
When using the Athletic feat, the grappler can drag the grappled creature another 25 feet. As it's speed at the time of dragging is 30ft, and 5ft was expended by standing up, 25 feet of movement are left to walk around with. If releasing grapple this would increase back to 55 feet.
Implications
Dashing
An interesting exception occurs when making use of the dash action.
The rules for dashing are as follows:
When you take the Dash action, you gain extra movement for the current turn. The increase equals your speed, after applying any modifiers. With a speed of 30 feet, for example, you can move up to 60 feet on your turn if you dash.
Any increase or decrease to your speed changes this additional movement by the same amount. If your speed of 30 feet is reduced to 15 feet, for instance, you can move up to 30 feet this turn if you dash.
(PHB, pg. 192, "Dash")
Curiously, (depending on your interpretation of the rules) this means that the grappler can use the dash action to gain 60 feet (as they are not (yet) invoking the grappling rule) of movement that is not affected by lowering your speed, as this added movement specifically allows you to ignore that threshold. (You would not be able to dash at all, otherwise)
However, the rules here provide enough language to imply that the movement gained from dashing is reduced by the grappling rule, as no timeframe is mentioned for "Any increase or decrease to your speed [...]" to take effect.
Thus we fall back to "The DM/GM has the final say". I recommend the second interpretation, as that is most likely RAI.
Why Phil S. is wrong
Though the reasoning of @Phil S. is sound, (except for equating movement to speed) it's their sources that are wrong. As a sharp eyed reader may have noticed, Phil S. refers to "Rules", which is either a book that does not exist, a ruleset that is outdated or a ruleset for a different game. If the ruleset is the Basic Rules, Phil S. may have an outdated copy. According to wikipedia, these rules have been subject to errata and have subsequently been updated to match the Player's Handbook.