Question says it all really. Does increasing dexterity in Elden Ring improve the cast speed of weapon arts?
No, it doesn't
Dexterity's effect information indicates that it "reduces the cast time of Spells" and it really is limited only to that. Of importance is that "Spells" encompasses both sorceries and incantations. Weapon arts are not considered spells, and thus are not affected by Dexterity's cast speed increase.
It's quite easy to get confused about this, though, especially when you consider the Sword of Night and Flame's Night and Flame stance skill, which can be perceived as a sorcery. It scales with Intelligence and appears to cast as a spell, but it's still considered a "skill", so its cast speed is unaffected by Dexterity.
According to Fextralife...
Dexterity also affects the cast speed of Sorceries and Incantations. Every point in Dexterity increases the cast speed slightly.
So yes, it does increase. The comments on that page claim that each point of dexterity increases casting speed by 0.5%, but this hasn't been confirmed.
Edit: I've completely misunderstood what you asked. If I'm correct, you were asking about weapon arts (which, I assume, Weapon skills), not sorcery or incantations. This answer, however, will be kept for future reference.
I don't know if Dexterity affects weapon skill's speed. In fact, I did not know that dexterity affected sorcery or incantations. However, judging from the page, it does not affect them, seeing as there is no mention of weapon skill casting/charging speed.
Plus, if it does, what would be the point of using a Strength-based build? PvE-wise, it's good, but in PvP, if weapon skills casting/charging speed are affected by dexterity, they will easily lose.