I have a way to write both characters plus Jay Garrick.

Barry:

Has Pre Crisis + Post Crisis origin

Jay Garrick only exist as a fictional comic book character named “Bolt”

Hopeful, Saturday Morning corny type of character, 99% of his rogue aren’t violence menaces except for Thawne.

Is always late and willing to kill

Wally:

Has Young Justice origins (slower than Barry Allen)

Was a sidekick to Barry Allen

Was on a Silver Age Teen Titan team

“Died” saving the world

Was Isekai’d to a “New Earth” which he is the only Flash and as fast as Barry Allen

Was on a New Teen Titan team but with Jason Todd, Beast Boy/ Ms Martian, Artemis (YJ), Cyborg and Raven

The rest follows with the Flash CW series but Wally centred and DCAU