Eyewitnesses tell us Meghan attempted to be as incognito as possible -- with 4 bodyguards in tow -- when she entered the VFS Global UK Visa Application Centre Thursday in Chicago. We're told she was there to finalize her application for a UK visa.

The whole thing only took about 10 minutes, and the entire office was cleared for her. We're told she was friendly with the staff. As for her disguise -- shades and a White Sox cap to blend in with locals.

Meghan forked over about $1,500, maybe a little more, for premium processing -- so she should get her visa within a few days ... according to our sources.

Based on the visa application criteria, she most likely submitted for a family visa, and applied as a partner or spouse. That visa requires her to confirm all previous marriages, and her plans to marry or become a civil partner within 6 months of arriving in the UK.

We can hear her now ... "Maybe you've heard of my fiance."

After 6 months, Meghan can go for permanent citizenship, which Kensington Palace has said she'll do. No fast track, though ... that process can take up to 3 years.