TL;DR: Approached the international help desk, emailed the New Delhi office at an address I found online. Not sure what worked, but they asked us to resubmit our documents, and we got the visa.
Gayot has provided an excellent guide about how to handle the situation. I approached the international help desk (Option 4).
When we first discovered the mistake, we called the VFS that processed our application and they said they are not able to handle requests for reviews. When we insisted that this was not a review per se, but rather a request for correction, they said that we could at best try to email the high commission. This was misleading, as the high commission doesn't respond to visa inquiries at all.
We found an email address for the New Delhi office from which we had previously received communication. We emailed that address outlining our situation. But we weren't sure if it was a monitored email address.
We then approached the international help desk. At the time the phone lines were not available, so we used the paid email service. After 24 hours, we decided to call them since we hadn't heard back. They listened to what we had to say, and asked us to send some more information via email, such as date of biometrics and some information that wasn't available when we were on the phone. When we did that, they sent an email stating that we were not eligible for a review.
I called them again and had to repeat the same series of events, and one agent said that they are going to escalate the issue.
I called back within 48 hours to check if there was an update, since the date of convocation was approaching quickly. When I asked about a timeline, they said it could take up to 15 business days to hear back.
I called back shortly after, because I got a sense that they didn't have a process around raising the urgency for rare issues such as mine. I spoke with a rep for 30 minutes, and after I insisted repeatedly and made my case to them, they said that they were going to arrange a review. I was a little puzzled at this, since the language previous reps used was "escalation" and not a "review". As we were finishing up details, the phone got disconnected - it happened at the 30 minute mark.
I was extremely frustrated at this point. I had already spent an inordinate amount of time waiting on the phone and spending 1.37 pounds per minute in addition to international calling charges. I called them again, and this time the rep said they were going to escalate it and it would take 15 business days.
The next day, I got an email asking me to go to the VFS with my passport and documents.
When we got to the VFS center, they seemed to have no clue about this - they asked us about our appointment and such. But we got them to accept our documents.
About a couple of weeks later, we called VFS and they said that our passports had arrived. We found out that we were issued visas.
**I will post the exact timeline of events later.