Oakley will cover the lead role of Jenna having performed the role on the North American tour.
Oakley, who's other credits in the States include Wicked and Chicago, will make her West End debut in the musical, performing for a week from today (Monday, January 13).
It follows cast illness which has seen current principle Lucie Jones as well as both understudies Olivia Moore and Sarah O'Connor forced to rest.
The production was forced to cancel both of its performances on Saturday (January 11).
The hit musical said it had taken the "difficult decision" due to "illnesses in our cast".
A statement posted to the show's official Twitter account read: "Due to illnesses in our cast, we made the difficult decision to cancel both performances today, Sat 11th January. Our doors will re-open on Monday Jan 13th.
"Please contact your point of purchase to arrange a new date to visit us. We can’t wait to have you down at the diner 💗"
However proving that the show must go on - even if not quite as intended - members of the musical's company staged an impromptu concert version of the show for those that had tickets.
Cast member David Hunter tweeted: "A strange day @waitresslondon made easier by our kindhearted company.
"A sincere thank you to our audiences for your understanding and bringing such high spirits to our little concert. x"
It was announced earlier this year that Waitress will close in London's West End on July 4 2020 before heading out on a first UK tour.
Before then, Sara Bareilles - who composed the musical's score - will appear in the cast for a limited time from January 27 together with Gavin Creel.
The pair, who both previously starred in the show on Broadway, will take on the roles of Jenna and Dr Pomatter through to March 7. Jones and Hunter will return to the roles of Jenna and Dr Pomatter on 9 March.