Just in terms of curriculum and intensity, what could one expect to be the differences between a c.a. 1900 London or other urban English primary school and a c.a. 1800 primary school? Were there any shifts in the weight of certain classes (like maybe a shift away or towards the study of economics, just spitballing) or were there any trends in education around the turn of the 19th century that did not follow through to the 20th? Apologies if this is too broad a question (it definitely is broad).
Well, it is a little late, but like you said, it is a broad question. Boys, girls, upper, middle, and lower class people were all educated differently. Both periods you specify are sort of transitional. I've been meaning to write this up, so thanks for the question.
1800:
Children under age 8 or so were usually educated privately at home. This hasn't changed much. The quality of this foundational education was dependent on the individual instructors. Wealthier children had governesses: live-in employees and were part-teacher, part-nanny. These women's own educations (and teaching abilities) were not assured to be good. They were sometimes supplemented by more credible tutors in academics and the arts. Poor people were either educated by their parents/older siblings, or at Dame Schools (much like governesses, but run out of the teacher's house: part-school, part-daycare). The other big source of early education for the poor was Sunday School. The Church of England's goal was to make people literate enough to read the Bible, and so children were taught to read with Christianity. Religious education was a major theme of all early childhood education. Most poor people and most girls continued these methods of private, semi-private, or self-education, beyond age 8.
Upper-class boys went off to the public boarding schools (Eton, Harrow, Winchester, etc.) around age 8 to begin a "Classical education." They focused on Latin (and to a lesser extent, Greek) grammars as a foundation for studying ancient history, politics, philosophy, and Christian doctrine. The goal was to turn these boys into society's leaders, with the assumption that most would be continuing on to university (where the science and math would come in). For the lower-class boys there were free charity schools, which had also begun centuries earlier as these same sorts of Latin "grammar schools." However, only the very brightest of these students continued on to higher education (via scholarship). Most of them were placed into apprenticeships by age 14 (the same premise as vocational school). As a result, the curriculum was changing in the 18th century. Latin was still taught (it was required to be under school charters), but there was more of an emphasis on English grammar and basic maths -- the sorts of things that would be more useful to future tradesmen. This shift to Reading, Writing, and Arithmetic was slowly taking over ca. 1800.
In addition, both types of schools began offering fee-based elective courses that filled the gap in terms of practical and social skills: Academics: modern languages (especially French), geography, modern history, geometry; Commercial: Bookkeeping; and Arts: music, dancing. (Those boys had to learn those 20-minute dances somewhere!)
Some fee-based schools did open with the new English-based practical, commercial curriculum. Many of the "charity schools" were admitting fee-paying students alongside the boys who attended for free. Thus, the growing need for middle-class education was being filled.
Girls' education was similar to the lower and middle-class boys - the emphasis on English grammar, literature, writing, and basic math. It was similarly practical: they focused on things like spelling, penmanship, and pretty phrasing in their writing. They learned math for use in household accountancy. They learned French, history, geography, and home crafts like drawing and sewing. In addition, girls' schools emphasized moral, Christian conduct, and the traditional deportment we so often associate with early women's education. Electives were in things like music, Italian, German, geometry, botany, even astronomy -- it varied, presumably depending on what teachers a school could get. There were charity schools for girls, but most of this is aimed at the middle-class set. Professional careers for educated women were thin on the ground - many girls attended school with a goal of becoming educators themselves; most were hoping to gain enough respectability and know-how to become useful wives. Some, like the boys, were entered into apprenticeships (most likely domestic service).
Upper-class girls were educated at home, and like their male counterparts, they were being groomed for roles at the top of society. Jane Austen's famous list of "accomplishments" (music, singing, drawing, dancing, and the modern languages) is not far off, but it is worth noting that some upper-class people did indulge their daughters with secondary-level academic tutors. Despite the fact that they were unlikely to get much mileage from this education, studying math, history, natural philosophies, and literature were wholesome occupations for young teenage girls -- and would allow them to conduct intelligent conversations with learned men.
Education, Economic Change and Society in England 1780-1870, by Michael Sanderson, 1983
Educating the Child in Enlightenment Britain: Beliefs, Cultures, Practices, 2009, esp. ch4: Learning and Virtue: English Grammar and the Eighteenth Century Girls' School, Carol Percy.
This blog post, which is slightly careless on denoting the direct quotations from his sources, but very well cited.
19th Century Changes: