There are two different things happening here: your movement speed being halved, and spending half your movement speed. These work differently, and the order matters. The end result is that the grappler can't move-drag after standing up from prone. Here's how it works:
You start with your full movement speed.
Let's use 60′ for the sake of example.
Standing up costs half your movement.
Your current speed is 60′, half of which is 30′. To stand up you spend 30′ of movement. Your speed is still 60′.
Dragging while grappling halves your movement speed. Your movement speed this round has now been reduced to exactly how much you've already spent, so you will have 0 feet of movement to spend.
Your current speed is 60′, but attempting to move-drag a grappled opponent changes it to 30′. You have already spent 30′ of movement and have zero feet left to spend.
The end result is that after standing up, you still have half your movement left, but as soon as you try to drag a grappled opponent you will have no movement left and remain where you are, so realistically you won't bother trying to drag that round. (You could still end the grapple and move your remaining half movement though, of course.)
Levitate doesn't give you a different Speed
The spell Levitate permits you to move yourself in a way you couldn't before. Specifically (PHB, p. 255, bold added):
You can change the target's altitude by up to 20 feet in either direction on your turn. If you are the target, you can move up or down as part of your move. Otherwise, you can use your action to move the target, which must remain within the spell's range.
This may sound like it gives you a "levitating speed" of 20 feet, but the spell does not specify that it does this. Contrast that with the spells Fly or Gaseous Form, which state:
Fly ...The target gains a flying speed of 60 feet for the duration. (PHB, p. 243)
Gaseous Form ...While in this form, the target’s only method of movement is a flying speed of 10 feet. (PHB, p. 244)
In the absence of similar text in the Levitate spell's description, we must conclude that you don't gain any new speed from Levitate. Instead, Levitate does exactly what it says it does: it allows you to move up to 20 feet in either direction "as part of your move."
What difference does that make?
So we've established you move via levitation "as part of your move." What does that mean practically? The rules on movement state (PHB, p. 190):
On your turn, you can move a distance up to your speed.
In your given example, a character who has walked 30 feet this turn (their speed) is under the influence of the Levitate spell (which they cast), and wants to immediately levitate upward. But since they've already moved "up to their speed" they cannot move further "as part of their move" this turn (unless they take the Dash action or otherwise give themselves additional movement). Thus, in your example, the character would be unable to rise, but not for the reason you suggested (having to do with multiple speeds).
Note that this can also work in your favor. If you moved 20 feet on the ground and then used your movement to change your altitude via Levitate, you could raise or lower yourself 10 feet with the spell (assuming your walking speed is 30). This would have been impossible if Levitate gave you a new speed (which, again, it doesn't), since you would subtract the 20 feet walked from the 20 feet "levitation speed" and find yourself unable to move via levitation at all that turn (due to the rules on moving with multiple speeds which you quoted in your question).
The difference between "speed" and "movement" is complicated, and this is one of the more sticky examples of them interacting in strange ways. But the bottom line is that you don't have to worry about the rules on multiple movement speeds when using this spell because the Levitate spell doesn't give you a different speed.
Best Answer
Rules As Intended (RAI) - Standing up costs movement equal to 1/2 your highest speed
Jeremy Crawford has put out an official clarification which says:
This seems like a logical and reasonable way to approach this. If you have 30' walk speed and 60' fly speed, the maximum distance you can move in one turn is 60'. You can split the proportion of walk/fly in many ways, but the end result will be 60' of movement or less. When looked at this way, a character's speed can be thought of as being effectively equal to the maximum type of speed they have.
Thus, it makes sense then that one would simply expend movement equal to 1/2 of that maximum speed (the character's effective speed) when standing up from prone.
So, in OP's example, upon standing up they would have 0 feet of walking and 30 feet of flying movement left.
You would subtract 30 from both because it is consistent with the way that using movement when you have multiple speeds works. If one had used 30' of either walking movement normally then switched to fly one would have the same result (0' walk/30' fly remaining). Same if you started with flying and switched to walking.
RAI vs RAW
Since this is a (now officially acknowledged) hole in the rules, if your table does not care about designer intent and instead wishes to follow the rules as they seem to have been written you can use the logic below. I think it makes a compelling argument that can be made to interpret the rules in the context only of what is written (not taking intent into account).
However, I, personally, do like to follow designer intent whenever possible especially if neither option has a significantly different impact on how much fun the game is (and especially if the RAI are more fun).
Following that logic below leads to a very similar end result as the RAI. In many cases they end up being the same amount of speed remaining, but the proportions of the speed leftover may differ.
Rules As Written (RAW) - Standing up costs movement equal to half of all of your speeds
One way to justify the rules from a purely RAW standpoint is as follows. Since this is clearly a rules hole, arguments could be made for other readings.
Prone's description says:
Jeremy Crawford has said:
Thus standing up from prone would cost movement equal to half all of your speeds.
Additional evidence can be found in the description of grappling
When the game says "speed" without any other qualifiers they are referring to your speeds in general.
Take the case of grappling as further evidence for the fact that "speed" refers to all of your speeds. The grappled condition says:
Clearly the intent here is to say the creature (outside of any extraordinary abilities) does not have any speed and cannot move as a result. If you assume speed here means only one speed, then a creature with a walk and fly speed would be able to fly away with no issues ignoring the grapple completely. This doesn't make sense, and it is clearly not the intent.
Luckily, Jeremy Crawford has once again clarified and again supports this reading here:
So clearly, when the game says speed they are referring to all of your speeds. Thus "an amount of movement equal to half your speed" is also referring to all of your speeds.
Thus, even other ruling reaffirm the above interpretation: standing up from prone would cost movement equal to half all of your speeds.
There are even other areas where this same use of speed occurs. See Does Longstrider apply to other speeds? and How do Boots of Speed affect other movement rates? for just two other examples.
Your example
In your specific example you would have 15 feet of walking and 30 feet of flying movement left after standing.
After standing up you could then walk 15 feet and fly 15 feet or you could just fly 30 feet (or some division between the two).
The math works out as well: before standing up (with 30' walking/60' flying) you had an effective maximum movement of 60' (since using multiple movements means that essentially you are capped at your largest speed). After standing up (15'/30') you have an effective maximum movement of 30', so half of the original.