It's very area-dependent. The Mint previously disclosed "release cities", though they haven't done this for the past few quarters. They also said that quarters would start showing up in circulation about 4-6 weeks after the official release date, given the time needed to channel the quarters through the armored car carriers and banks. https://www.coinworld.com/news/prec...arter-dollar
However, getting those coveted uncirculated rolls seems much more nuanced than showing up at a release city 4-6 weeks after release date. Based on YouTube CRH videos, eBay scouting, Coin World
reports, and the PCGS
Bounty, W quarters show up right away, and sometimes even before the official release date. They also show up in other places. I live 250 miles from the closest release city, but a really nice batch of War in the Pacific quarters showed up in October, 2019, 4 months after release date. It was an amazing run, and I got a decent amount of shiny new W quarters out of that batch; banks had new rolls for around 6 weeks. Since then, smaller groups of River of No Return and Weir Farm quarters have surfaced, but nothing as epic as the WIP's. A lot also depends on the setup in your area. Some places utilize one armored car company (Loomis or Brinks or Garda), while bigger cities will have more than one. Based on my limited experience (and watching YouTube videos), each company seems to have different distribution schemes - people tend to have more luck with Loomis rolls, although other rolls have also been successful.
Location is thus key as well, I'd even say more important than timing. I've tried a few tricks, like looking at eBay listings to see where the new W quarters are coming from. Sometimes, a modest drive can pay off handsomely, as some places are indeed W cold zones.
Asking around can also be very helpful. Many bank tellers are super-friendly and will go out of their way to look for uncirculated rolls.
Circulated rolls also have W's, about 1-2 per $500 box in my area, though again, that will be largely area-dependent, as they are recycled from quarters that were previously circulating there. In my experience, bank wrapped circulated rolls still have better odds for W's compared to customer-wrapped rolls, as the latter have been sitting around in a jar since before W's came out or may be tailings from someone else's search.
Good luck in your quest! May you find those W's!