Since Jeremy Crawford is the rules authority and has said otherwise, this answer is Wrong™. However, I think this interpretation follows pretty clearly from the rules as written and consider his message an update/clarification which effectively overrides the plain text. That's okay. (Particularly because at least in this specific case of a mere +10 to one creature, I can't imagine this being significantly game-altering either way.)
Speed without any qualifier refers to walking speed. Longstrider increases just that speed, not any other movement modes a creature may have. This is part of 5E's "rules do what they say they do"-based design.
From the Basic Rules:
Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round.
(And same in the Player's Handbook. Emphasis added.)
And from the Monster Manual or Basic Rules on Monsters:
All creatures have a walking speed, simply called the monster’s speed. Creatures that have no form of ground-based locomotion have a walking speed of 0 feet.
Some creatures have one or more of the following additional movement modes.
... and then it lists Burrow, Climb, Fly, and Swim and describes the different rules for each of those.
And from a random monster's stat block:
Speed 20 ft., fly 120 ft.
In the rules, by definition, the term speed by itself refers to walking movement. When a different movement mode is intended, it is referenced specifically. The term "walking speed" is used simply for clarity in cases where multiple modes of movement are discussed. There's no reason to draw from these examples an overturning of the general definition.
From an "exception-based design" perspective, the general rule is that "speed" limits the distance you can walk in one round. Other movement modes are granted by special abilities, and there are specific rules for how each of these works. Since Longstrider does not refer to these specific other movement modes, it does not normally apply to them.
In some cases, though, the feature which grants another movement mode refers to your regular walking speed. The barbarian Totemic Attunement and cleric Stormborn features are examples of this, because both of them use the wording "you have a flying speed equal to your current walking speed". Here, increasing your speed does affect your flying speed, because the specific rule says it does. The Spider Climb spell is an example of this same type of feature for climbing.
The flying speed granted by Potion of Flying or Winged Boots would also be affected (same language referencing your walking speed) but that from the Fly spell would not (because it grants a flat flying speed of 60 feet).
You might think "but wait, can't you Dash with your other movement modes"? You can, but that's because Dash doesn't increase your speed. It gives you extra movement — it's an exception to the normal rule that your movement is limited to a distance equal to your speed. Arguably, the creature above only gets 20 more feet of movement from Dash even when flying, and Longstrider would extend that to 30. (But I've never seen anyone play that way, and Mike Mearls agrees with someone asking if Dash plus the Fly spell would allow 120 feet of movement.)