St Botolph's may have originally been called the National School.
Kelly's 1900 Lincolnshire directory says
"National (School), Pump Square (mixed), established in 1815 and rebuilt in 1849 in the Elizabethan style for 470 children; average attendance 402; Alfred James Clark, master; Miss Gertrude Teamby, infants' mistress."
In Kelly's 1926 Lincolnshire, the school is described as 'Church of England.... H G Woods, master; Miss A Lewsley, infants' mistress'.
Not connected with the original query, but also possibly of interest is that Launchbury's cycle shop was previously the United Methodist Free Sunday School, established 1872. Stone tablet above the shop sign says it.
The Free Methodist Chapel itself, first erected in 1856, was re-fitted and had a new organ in 1906, and was then able to seat 450 people.