There is no timer. This is from the FAQ located here: https://support.riotgames.com/hc/en-us/articles/115008474148-Honor-FAQ
LOADING SCREEN FLAIRS
How do I get a loading screen flair?
First, hit level 3. Then earn honor from two premade teammates or one stranger in one game. The flair will unlock for your next matchmade game.
My flair disappeared after one game, why?
They’re meant to show off that you’re coming off a great game, so they only last one game. Once you're level 3 or better, get an Honor from two premade teammates or one stranger to unlock flair in your next game.
Is there a cooldown on loading screen flairs?
Nope. If you keep earning honors, you'll keep your flair across each game you rock the Rift and earn honors.
Will I get a different loading screen flair per category?
Not today, but that's a damn good idea.
Do they get bigger like Champ Mastery?
They do! At level 4 and 5 the flairs grow.