Yep, you'll usually show up in search engines after a little while.
How long it takes depends a bit on where else you've linked your band.bio (like on Instagram, etc.), how unique your name is, and exactly what people are searching for.
If someone searches for "BANDNAME + band" for example - which, according to our Google keyword data, happens quite often - then your chances are looking really good.
Here are a few examples:
https://www.google.com/search?q=wbr+band
https://www.google.com/search?q=Lakestaff+band
https://www.google.com/search?q=sonrise+band
https://www.google.com/search?q=dont+ask+band
It often works even without adding "band," provided your name isn't too ambiguous:
https://www.google.com/search?q=sonrise
https://www.google.com/search?q=Lakestaff
Keep in mind that search engine results depend on location, language, and search history. So, your exact ranking can vary a bit based on those factors.
Redirecting your own domain to your band.bio can also boost visibility, especially if that domain already had some traction. Ideally, you should use a direct, server-side 301 redirect. The second-best option is a 0-second meta refresh (<meta http-equiv="refresh" content="0; url=...">) if you can only upload an HTML file.
We also automatically set page titles, descriptions, and preview images using your band name, genre, and location so that search engines can categorize your page correctly.