Benefit of Internship Referrals & Application Timeline

I was thankfully able to get a referral for multiple internship positions through networking. Other than simply having the referral marking, does receiving this referral benefit my application? I've heard that it cuts through the ATS and goes straight to the recruiters, which sounds great. I'm also curious as to how soon y'all were contacted after applying and what the timeline looks like. It's been about a week since I applied, and I'm interested in how long I should expect to wait if they end up considering me a good candidate. Thank you for the help!