The pair departed for the day of engagements on Wednesday evening, journeying north on board the royal train.
Crowds -- including swarms of excited schoolchildren waving flags -- gathered as they arrived at train station in Runcorn, where they were greeted by David Briggs, the Lord-Lieutenant of Cheshire.
The day is likely to serve as a masterclass in royal etiquette for Meghan, who will get to see how Britain's longest-reigning monarch navigates these more mundane events, which are a staple part of life as a working royal.
While visiting the bridge, the royals were introduced to architects, planners and community representatives before watching a performance by local schoolchildren. The Queen then unveiled a plaque at the site to mark the opening of the bridge.
Later, the Queen and her new ganddaughter-in-law will head into the city of Chester where they will open Storyhouse, a new library, theater and cinema. During the visit, they will watch a performance from a dance company for people in recovery from addiction.
Following that, Meghan -- under the watchful eye of the monarch -- will attend a lunch at Chester Town Hall.