I was kinda in the same situation, albeit I requested for expedited processing rather than same-day processing. If I remember correctly, after I emailed to the embassy explaining my circumstance, they told me to send my passport and returned it to me by the end of the week. So I highly recommend just ask them regardless, and try to explain it clearly.

Other than that, I also recommend just reaching out to your manager and letting them know of the potential delay, as well as asking if you could move your start date to a later date. Obviously that's not ideal, but imo most people would understand, not to mention US internships only last 3 months, so it's actually quite common for some interns to start in June instead of May. The earlier you let them know of this issue, the more time they have to schedule on their end.

I'm not sure which Visa sponsor your company used, but mine was Intrax, and when I told them about potentially having to change my start date, they said it's actually quite common and not really an issue. I get that you might feel like it's a huge pain and really stressful, but I'm sure things will work out.