The OS 25 inch map published 1907 has schools (plural) marked here also "St Leonard's Ch. (site of)".
Ruddock's Directory of the City of Lincoln 1919 has two schools listed :-
"Miss COOKSON's GIRLS' SCHOOL, Greetwellgate Miss S. Brown"
"EASTGATE INFANTS Miss M. Russell"
This image seems to be of what was the girls school. School is listed and Historic England has further information :-
"Formerly known as: Eastgate Primary School GREETWELL GATE. School with attached teacher's house and boundary wall. Founded 1850 by a Miss Cookson, rebuilt 1881-82 by William Watkins of Lincoln"
DB 7 August 2020
Lincoln, St Peter In Eastgate Infant School, Miss COOKSON's GIRLS' SCHOOL, William Watkins