How to choose a delivery city on the main page in the header?

Question

WordPress

The essence of the problem: in the checkout, they made a choice of the delivery city, and wrote their own cost for each city.
http://prntscr.com/nri21i
and now we want to make the same city selection line http://prntscr.com/nri2m6appear in the header on the main page in something like this http://prntscr.com/nri3mu
. And when the client selects the delivery city on the main page, the information with the selected city should, in theory, be pulled into the checkout.
I understand that somewhere you need to tear out a piece of code that displays a string with a city selection .. but I don’t know where to find it.
Or I'm wrong?
Tell me how to do it?