Woah

That was a find Rosie, great stuff

Annie Elizabeth Yates was born 25 July 1910, and died 1995 in Leeds under same name - looks as if never married. Her birth date on Death Index matches the 1939 entry. I think she is likley to be the "Jennie Elizabeth" you found?
Haven't found Harry yet- but here is Mary Elizabeth going to the US in 1909, it says her mother is M Brannan. All fits with your findings!
TRAVEL 1909: SS Cedric arrived NewYork 28 Aug 1909, out of Liverpool 21 Aug 1909Line 28: Mary E BRANNAN 23, single, Cigar maker,
Last residence in UK: Leeds, England
Relative in UK: Mother M Brannan, 2 Back Sackett Street, Leeds.
Destination: Chicago Illinois
Going to in US: (illegible) U C Yates, 423 No. Clark St, Chicago
Description: 5',1", Fair sking, Dark Hair, Dark eyes. Born York, England
And pretty sure this them in 1910 - Harry was a Bricklayer in the UK 1911:
CENSUS 1910: 310 Tell Court Street, Chicago (Ward 23), Cook, IllinoisHead: Harry YATES 25, married 1, b England, parents b England, imm. 1905 Pa. Bricklayer (Building). Married 1 year.
Wife: Elizabeth YATES 24, married 1, b England, parents b England, imm. 1909. No children born of marriage.
Lodger: Zach? Haines or Maines, 37, single, b England, parents b England, imm. 1909 Al. Shoemaker (Factor
So possibly married in Chicago also?
Cheers
AMBLY