There is no company that provides the courier service - VFS Global is just a website where you can purchase services offered by UKVI.
Yes, thank you, I knew I had the wrong phrase but couldn't think of how they actually worked together!
Okay so upgrading to Priority worked then, it looks like?
I hadn't had my coffee yet, so here's the timeline and hopefully it's more clear:
-Late June: Mailed 3 non-priority applications to Sheffield. One as wife of a citizen, and two dependent children. We meet the financial qualifications through cash savings.
-Late June: FedEx shows it's been signed for
-July 7: Email acknowledging each application and one asking for a return slip that is not FedEx. Go to VFS Global, purchase a return carrier for each application (they only requested it regarding my application, but VFS says you need to purchase one return per application, so I just did all three) and email it to Sheffield with the VAF number as they requested. Then I went on these forums, saw people had been waiting since May, freaked out, went back to VFS Global, purchased priority upgrades for all three.
-July 8: Wake up to an email from Sheffield requesting birth certificates for both children (I could not find anything saying we should include it the first time and thought that was strange. Glad they followed up instead of just rejecting us right away!!)
Will keep you all updated to see if the late upgrade to priority makes a difference. Though I've also read that cash savings can go quicker because there's less to verify. We also included my husband's job offer letter sowing he's to start in September, and we included with the priority upgrade that I'd really like the kids to be there at the start of the school year, so some pity might be involved as well. Or, it could still take 12 weeks and there's nothing to be done. At least this way I feel like I've done all I can.