This is not so straightforward, see Peirce, Frege, the Logic of Relations, and Church's Theorem by Dipert for a sketch of history. The notation Russell used was not created by Peano, and certainly not by Frege. Nobody used Frege's convoluted notation except his own Begriffsschrift (1879), not even Frege himself afterwards. Peano did contribute, but the basis of Principia's notation and logical content was actually Schröder's Vorlesungen über die Algebra der Logik (1895), Principia's precursor in symbolic treatment of modern logic. In its turn, Schröder's treatment goes back not to Frege but to Peirce, who invented predicate logic with quantifiers in 1881-85 independently of Frege. Here is Dipert's surmise:
"Neither Peirce nor
Schröder had the services of such an excellent propagandist as Russell. The Peirce-
Schröder calculus was portrayed as purely algebraic, without the variable-binding
operators Peirce regarded as essential and to which Schröder usually resorted; its
weaknesses were rhetorically exploited with the bon mot 'too complicated'; its
subtlest achievements were ignored (e.g., clever theorems proven and Peirce's
insights into the differences between the monadic and polyadic predicate logic); and,
in a final injustice, the development of the theory of relations in Principia Mathematica
owes most, especially in notation, to Schröder via the influence of Peano rather
than to Frege, but it was presented without substantial acknowledgement.
Alfred Tarski is one of the few logicians or historians writing in the 20th century
who seems to realize the proportions of this injustice."
The Peirce-Schröder's notation was more mathematician friendly and structurally closer to the modern one than Russell's. They used $\Sigma$ and $\Pi$ for the existential and universal quantifiers, and did not use dotting for punctuation. That was Peano's creation, see SEP, Use of Dots for Punctuation. He also changed $\Sigma$ to $\exists$ for the existential quantifier, and $(x)$ for the universal was Russell's preference. Hilbert-Ackerman's Grundzüge der Theoretischen Logik (1928), the first textbook on mathematical logic, used Principia's notation, but the dotting is already gone in Hilbert-Bernays's Grundlagen Der Mathematik (1934).
Gentzen proposed $\forall x$ in 1935, by analogy to $\exists$, but Principia's version and the dotting persisted well into 1950-s. Church, Gödel, Quine used it, and so did Rosser's Logic for Mathematicians (1953). However, Quine's From a Logical Point of View (1953) already uses parentheses instead of dots. The switch is likely associated with the "transfer of ownership" of logic from philosophers to mathematicians in the late 1950s early 1960s. The catalyst might have been Bourbaki's systematization of mathematics in Éléments de Mathématique. Summary of the first volume, Théorie des Ensembles (Fascicule de Résultats), came out in 1939, and the chapters on formal logic and set theory in 1954. Bourbaki used Gentzen's quantifier and no dotting. Mendelsohn's Introduction to Mathematical Logic (1964), which became a standard textbook, already used modern notation.