Okay, so First off: hello, I am new here.
I won a free game from EpicBattleAxe, I got my code last night.
When I try to input the code though, it says it expired.

I will, thanks

It may also be worth talking to the folks who gave you the code, if you haven't tried that already. Perhaps they made a mistake?

anyone know how long it usually takes to get these replies?
This was kind of the highlight of my week

First, they are in Europe, so here it's 4 AM now.
Second, the last support ticket I sent was replied after two days. So if the same is for yours, you get a reply in around 32 hours.
Can't help more here, sorry.

They usually respond within 24-48 hours of when a support request is received, but like Protoss pointed out, GOG is based in Europe, so if your request went in after the close of business in Poland, then they won't receive it until the next business day. Additionally, GOG was recently having a problem with their ticketing system that caused a bunch of new tickets to be automatically marked as solved before they were reviewed. That was supposedly resolved, but if you don't hear anything by Friday afternoon (your time), then you might want to send a follow up.