Hi Allie!
We have definitely followed similar paths in our journey across the pond!

We were also married in Vegas! The marriage certificate process was incredibly easy, and we were able to obtain our marriage certificate quite quickly after the wedding ceremony.
Where do you live in Chicago? I lived in Lakeview and Roscoe Village. I really miss my friends and the friendly faces I would see every day, but I won't miss the freezing cold temperatures. Oh, and I miss the Chicago Cubs too, although they are awful this year.
Anyway, I wish you the best of luck with your wedding and the visa process! As an FYI, my visa was granted within 14 business days, which was surprisingly quick given the current backlog. You should definitely be in the UK by Valentine's Day (and even by Christmas if you're aiming for that).
Please keep in touch and good luck!!!